REAL is set
NAT is epsilon-transitive epsilon-connected ordinal non empty non trivial non finite cardinal limit_cardinal countable denumerable non empty-membered Element of bool REAL
bool REAL is non empty cup-closed diff-closed preBoolean set
RAT is set
{} is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued set
the Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued set is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued set
NAT is epsilon-transitive epsilon-connected ordinal non empty non trivial non finite cardinal limit_cardinal countable denumerable non empty-membered set
bool NAT is non empty non trivial cup-closed diff-closed preBoolean non finite non empty-membered set
bool NAT is non empty non trivial cup-closed diff-closed preBoolean non finite non empty-membered set
Fin NAT is non empty cup-closed diff-closed preBoolean set
COMPLEX is set
INT is set
1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
{{},1} is non empty finite V40() countable set
[:REAL,REAL:] is Relation-like set
bool [:REAL,REAL:] is non empty cup-closed diff-closed preBoolean set
K462() is set
bool K462() is non empty cup-closed diff-closed preBoolean set
K463() is Element of bool K462()
[:COMPLEX,COMPLEX:] is Relation-like set
bool [:COMPLEX,COMPLEX:] is non empty cup-closed diff-closed preBoolean set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty cup-closed diff-closed preBoolean set
[:[:REAL,REAL:],REAL:] is Relation-like set
bool [:[:REAL,REAL:],REAL:] is non empty cup-closed diff-closed preBoolean set
[:RAT,RAT:] is Relation-like set
bool [:RAT,RAT:] is non empty cup-closed diff-closed preBoolean set
[:[:RAT,RAT:],RAT:] is Relation-like set
bool [:[:RAT,RAT:],RAT:] is non empty cup-closed diff-closed preBoolean set
[:INT,INT:] is Relation-like set
bool [:INT,INT:] is non empty cup-closed diff-closed preBoolean set
[:[:INT,INT:],INT:] is Relation-like set
bool [:[:INT,INT:],INT:] is non empty cup-closed diff-closed preBoolean set
[:NAT,NAT:] is Relation-like non empty non trivial non finite non empty-membered set
[:[:NAT,NAT:],NAT:] is Relation-like non empty non trivial non finite non empty-membered set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial cup-closed diff-closed preBoolean non finite non empty-membered set
K576() is set
2 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
3 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued Element of NAT
Seg 1 is non empty trivial finite 1 -element countable Element of bool NAT
{1} is non empty trivial finite V40() 1 -element countable with_non-empty_elements non empty-membered set
Seg 2 is non empty finite 2 -element countable Element of bool NAT
{1,2} is non empty finite V40() countable with_non-empty_elements non empty-membered set
dom {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V77() set
rng {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty trivial finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued increasing decreasing non-decreasing non-increasing V72() V73() V74() V75() V77() with_non-empty_elements set
<*> REAL is Relation-like non-empty empty-yielding NAT -defined REAL -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued FinSequence of REAL
K430((<*> REAL)) is V55() real ext-real Element of REAL
{{}} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
addnat is Relation-like [:NAT,NAT:] -defined NAT -valued Function-like non empty total quasi_total commutative associative having_a_unity complex-valued ext-real-valued real-valued natural-valued Element of bool [:[:NAT,NAT:],NAT:]
multnat is Relation-like [:NAT,NAT:] -defined NAT -valued Function-like non empty total quasi_total commutative associative having_a_unity complex-valued ext-real-valued real-valued natural-valued Element of bool [:[:NAT,NAT:],NAT:]
the_unity_wrt multnat is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
M is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
D ^ M is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
D ^ M is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
<*> k is Relation-like non-empty empty-yielding NAT -defined k -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued FinSequence of k
k is set
<*> k is Relation-like non-empty empty-yielding NAT -defined k -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued FinSequence of k
k * is functional non empty FinSequence-membered FinSequenceSet of k
k is non empty set
D is Element of k
<*D*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like set
k * is functional non empty FinSequence-membered FinSequenceSet of k
<*D*> is Relation-like NAT -defined k -valued Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like FinSequence of k
M is Element of k
<*D,M*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like set
<*D,M*> is Relation-like NAT -defined k -valued Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like FinSequence of k
<*D*> is Relation-like NAT -defined k -valued Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like FinSequence of k
<*M*> is Relation-like NAT -defined k -valued Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like FinSequence of k
<*D*> ^ <*M*> is Relation-like NAT -defined k -valued Function-like non empty finite 1 + 1 -element countable FinSequence-like FinSubsequence-like FinSequence of k
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
[:(k *),(k *):] is Relation-like non empty set
[:[:(k *),(k *):],(k *):] is Relation-like non empty set
bool [:[:(k *),(k *):],(k *):] is non empty cup-closed diff-closed preBoolean set
D is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence of k *
M is Relation-like [:(k *),(k *):] -defined k * -valued Function-like non empty total quasi_total Function-yielding V50() Element of bool [:[:(k *),(k *):],(k *):]
M "**" D is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
i is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
M . (x,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[x,i] is set
{x,i} is functional non empty finite V40() countable set
{x} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
{{x,i},{x}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
M . [x,i] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(k,x,i) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
M is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
i is Relation-like [:(k *),(k *):] -defined k * -valued Function-like non empty total quasi_total Function-yielding V50() Element of bool [:[:(k *),(k *):],(k *):]
i "**" D is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
Fg is Relation-like [:(k *),(k *):] -defined k * -valued Function-like non empty total quasi_total Function-yielding V50() Element of bool [:[:(k *),(k *):],(k *):]
Fg "**" D is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
db is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
k1 is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
i . (db,k1) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[db,k1] is set
{db,k1} is functional non empty finite V40() countable set
{db} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
{{db,k1},{db}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
i . [db,k1] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(k,db,k1) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
Fg . (db,k1) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
Fg . [db,k1] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
((k *),D) is Relation-like NAT -defined k * -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like Element of (k *) *
(k *) * is functional non empty FinSequence-membered FinSequenceSet of k *
(k,((k *),D)) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[:(k *),(k *):] is Relation-like non empty set
[:[:(k *),(k *):],(k *):] is Relation-like non empty set
bool [:[:(k *),(k *):],(k *):] is non empty cup-closed diff-closed preBoolean set
M is Relation-like [:(k *),(k *):] -defined k * -valued Function-like non empty total quasi_total Function-yielding V50() Element of bool [:[:(k *),(k *):],(k *):]
M "**" ((k *),D) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
((k *)) is Relation-like non-empty empty-yielding NAT -defined k * -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued Element of (k *) *
(k *) * is functional non empty FinSequence-membered FinSequenceSet of k *
(k,((k *))) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k) is Relation-like non-empty empty-yielding NAT -defined k -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued Element of k *
[:(k *),(k *):] is Relation-like non empty set
[:[:(k *),(k *):],(k *):] is Relation-like non empty set
bool [:[:(k *),(k *):],(k *):] is non empty cup-closed diff-closed preBoolean set
D is Relation-like [:(k *),(k *):] -defined k * -valued Function-like non empty total quasi_total Function-yielding V50() Element of bool [:[:(k *),(k *):],(k *):]
D "**" ((k *)) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
D . ({},x) is set
[{},x] is set
{{},x} is functional non empty finite V40() countable set
{{{},x},{{}}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
D . [{},x] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
{} ^ x is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
D . (x,{}) is set
[x,{}] is set
{x,{}} is functional non empty finite V40() countable set
{x} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
{{x,{}},{x}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
D . [x,{}] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
x ^ {} is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
M is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
the_unity_wrt D is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence of k *
M is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence of k *
D ^ M is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence of k *
(k,(D ^ M)) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,D) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,(k,D),(k,M)) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[:(k *),(k *):] is Relation-like non empty set
[:[:(k *),(k *):],(k *):] is Relation-like non empty set
bool [:[:(k *),(k *):],(k *):] is non empty cup-closed diff-closed preBoolean set
x is Relation-like [:(k *),(k *):] -defined k * -valued Function-like non empty total quasi_total Function-yielding V50() Element of bool [:[:(k *),(k *):],(k *):]
x "**" (D ^ M) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
i is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
Fg is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
db is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x . (Fg,db) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[Fg,db] is set
{Fg,db} is functional non empty finite V40() countable set
{Fg} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
{{Fg,db},{Fg}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
x . [Fg,db] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
x . (i,(x . (Fg,db))) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[i,(x . (Fg,db))] is set
{i,(x . (Fg,db))} is functional non empty finite V40() countable set
{i} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
{{i,(x . (Fg,db))},{i}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
x . [i,(x . (Fg,db))] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(k,i,(x . (Fg,db))) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,Fg,db) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,i,(k,Fg,db)) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,i,Fg) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,(k,i,Fg),db) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x . (i,Fg) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[i,Fg] is set
{i,Fg} is functional non empty finite V40() countable set
{{i,Fg},{i}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
x . [i,Fg] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(k,(x . (i,Fg)),db) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x . ((x . (i,Fg)),db) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[(x . (i,Fg)),db] is set
{(x . (i,Fg)),db} is functional non empty finite V40() countable set
{(x . (i,Fg))} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
{{(x . (i,Fg)),db},{(x . (i,Fg))}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
x . [(x . (i,Fg)),db] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
Fg is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x . ({},Fg) is set
[{},Fg] is set
{{},Fg} is functional non empty finite V40() countable set
{{{},Fg},{{}}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
x . [{},Fg] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
{} ^ Fg is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
x . (Fg,{}) is set
[Fg,{}] is set
{Fg,{}} is functional non empty finite V40() countable set
{Fg} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
{{Fg,{}},{Fg}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
x . [Fg,{}] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
Fg ^ {} is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
i is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
len D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x "**" D is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x "**" M is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x . ((x "**" D),(x "**" M)) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[(x "**" D),(x "**" M)] is set
{(x "**" D),(x "**" M)} is functional non empty finite V40() countable set
{(x "**" D)} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
{{(x "**" D),(x "**" M)},{(x "**" D)}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
x . [(x "**" D),(x "**" M)] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(k,(x "**" D),(x "**" M)) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,(k,D),(x "**" M)) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
M is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
((k *),D,M) is Relation-like NAT -defined k * -valued Function-like non empty finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like Element of (k *) *
(k *) * is functional non empty FinSequence-membered FinSequenceSet of k *
(k,((k *),D,M)) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,D,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[:(k *),(k *):] is Relation-like non empty set
[:[:(k *),(k *):],(k *):] is Relation-like non empty set
bool [:[:(k *),(k *):],(k *):] is non empty cup-closed diff-closed preBoolean set
x is Relation-like [:(k *),(k *):] -defined k * -valued Function-like non empty total quasi_total Function-yielding V50() Element of bool [:[:(k *),(k *):],(k *):]
x "**" ((k *),D,M) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x . (D,M) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[D,M] is set
{D,M} is functional non empty finite V40() countable set
{D} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
{{D,M},{D}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
x . [D,M] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
M is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
<*D,M,x*> is Relation-like NAT -defined k * -valued Function-like non empty finite 3 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence of k *
(k,<*D,M,x*>) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,D,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,(k,D,M),x) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[:(k *),(k *):] is Relation-like non empty set
[:[:(k *),(k *):],(k *):] is Relation-like non empty set
bool [:[:(k *),(k *):],(k *):] is non empty cup-closed diff-closed preBoolean set
i is Relation-like [:(k *),(k *):] -defined k * -valued Function-like non empty total quasi_total Function-yielding V50() Element of bool [:[:(k *),(k *):],(k *):]
i "**" <*D,M,x*> is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
i . (D,M) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[D,M] is set
{D,M} is functional non empty finite V40() countable set
{D} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
{{D,M},{D}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
i . [D,M] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
i . ((i . (D,M)),x) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
[(i . (D,M)),x] is set
{(i . (D,M)),x} is functional non empty finite V40() countable set
{(i . (D,M))} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
{{(i . (D,M)),x},{(i . (D,M))}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
i . [(i . (D,M)),x] is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(k,(i . (D,M)),x) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence of k *
M is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence of k *
(k,D) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence of k *
D ^ x is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence of k *
(k,x) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,(k,D),(k,x)) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
F1() is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
D + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
k + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
Seg (k + 1) is non empty finite k + 1 -element countable Element of bool NAT
k is set
{} |_2 k is Relation-like set
[:k,k:] is Relation-like set
{} /\ [:k,k:] is Relation-like NAT -defined RAT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
k is set
Fin k is non empty cup-closed diff-closed preBoolean set
bool (Fin k) is non empty cup-closed diff-closed preBoolean set
k is non empty set
Fin k is non empty cup-closed diff-closed preBoolean set
bool (Fin k) is non empty cup-closed diff-closed preBoolean set
the Element of k is Element of k
{ the Element of k} is non empty trivial finite 1 -element countable Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
{{ the Element of k}} is non empty trivial finite V40() 1 -element countable with_non-empty_elements non empty-membered Element of bool (bool k)
bool (bool k) is non empty cup-closed diff-closed preBoolean set
M is Element of bool (Fin k)
k is non empty set
Fin k is non empty cup-closed diff-closed preBoolean set
bool (Fin k) is non empty cup-closed diff-closed preBoolean set
D is non empty with_non-empty_elements non empty-membered Element of bool (Fin k)
the non empty Element of D is non empty Element of D
k is non empty set
Fin k is non empty cup-closed diff-closed preBoolean set
bool (Fin k) is non empty cup-closed diff-closed preBoolean set
the Element of k is Element of k
{ the Element of k} is non empty trivial finite 1 -element countable Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
{{ the Element of k}} is non empty trivial finite V40() 1 -element countable with_non-empty_elements non empty-membered Element of bool (bool k)
bool (bool k) is non empty cup-closed diff-closed preBoolean set
M is Element of bool (Fin k)
k is non empty set
[:k,k:] is Relation-like non empty set
bool [:k,k:] is non empty cup-closed diff-closed preBoolean set
bool k is non empty cup-closed diff-closed preBoolean set
D is Relation-like k -defined k -valued total quasi_total reflexive antisymmetric transitive Element of bool [:k,k:]
M is Element of bool k
D |_2 M is Relation-like set
[:M,M:] is Relation-like set
D /\ [:M,M:] is Relation-like k -defined k -valued set
bool [:M,M:] is non empty cup-closed diff-closed preBoolean set
F1() is set
bool F1() is non empty cup-closed diff-closed preBoolean set
F2() is Element of bool F1()
{} F1() is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued Element of bool F1()
bool F2() is non empty cup-closed diff-closed preBoolean Element of bool (bool F2())
bool F2() is non empty cup-closed diff-closed preBoolean set
bool (bool F2()) is non empty cup-closed diff-closed preBoolean set
k is set
D is set
D is Element of bool (bool F2())
M is set
F2() \ M is Element of bool F1()
the Element of F2() \ M is Element of F2() \ M
{ the Element of F2() \ M} is non empty trivial finite 1 -element countable set
M \/ { the Element of F2() \ M} is non empty set
i is set
x is set
k is non empty set
Fin k is non empty cup-closed diff-closed preBoolean set
bool (Fin k) is non empty cup-closed diff-closed preBoolean set
D is non empty non empty-membered Element of bool (Fin k)
M is non empty set
x is Element of D
k is finite countable set
D is finite countable set
card k is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
card D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
bool D is non empty cup-closed diff-closed preBoolean finite V40() countable set
M is finite countable Element of bool D
incl M is Relation-like M -defined D -valued M -valued Function-like one-to-one total quasi_total finite countable reflexive symmetric antisymmetric transitive Element of bool [:M,D:]
[:M,D:] is Relation-like finite countable set
bool [:M,D:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
rng (incl M) is finite countable Element of bool D
k is set
bool k is non empty cup-closed diff-closed preBoolean set
[:k,k:] is Relation-like set
bool [:k,k:] is non empty cup-closed diff-closed preBoolean set
M is Relation-like k -defined k -valued total quasi_total reflexive antisymmetric transitive Element of bool [:k,k:]
D is finite countable Element of bool k
(k) is Relation-like non-empty empty-yielding NAT -defined k -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued Element of k *
k * is functional non empty FinSequence-membered FinSequenceSet of k
rng (k) is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty trivial finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued increasing decreasing non-decreasing non-increasing V72() V73() V74() V77() with_non-empty_elements Element of bool k
dom (k) is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V77() Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
(k) /. x is Element of k
(k) /. i is Element of k
[((k) /. x),((k) /. i)] is set
{((k) /. x),((k) /. i)} is non empty finite countable set
{((k) /. x)} is non empty trivial finite 1 -element countable set
{{((k) /. x),((k) /. i)},{((k) /. x)}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
x is non empty set
bool x is non empty cup-closed diff-closed preBoolean set
[:x,x:] is Relation-like non empty set
bool [:x,x:] is non empty cup-closed diff-closed preBoolean set
i is non empty finite countable Element of bool x
Fg is Relation-like x -defined x -valued total quasi_total reflexive antisymmetric transitive Element of bool [:x,x:]
{ H1(b1) where b1 is Element of i : ( b1 <=_ Fg,a1 & not b1 = a1 ) } is set
db is Element of i
{ H1(b1) where b1 is Element of i : ( b1 <=_ Fg,db & not b1 = db ) } is set
card H2(db) is epsilon-transitive epsilon-connected ordinal cardinal set
(card H2(db)) +^ 1 is epsilon-transitive epsilon-connected ordinal set
{ H1(b1) where b1 is Element of i : S1[b1] } is set
k1 is finite countable set
card k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k2 +^ 1 is epsilon-transitive epsilon-connected ordinal set
k2 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
[:i,NAT:] is Relation-like non empty non trivial non finite non empty-membered set
bool [:i,NAT:] is non empty non trivial cup-closed diff-closed preBoolean non finite non empty-membered set
db is Relation-like i -defined NAT -valued Function-like non empty total quasi_total finite countable complex-valued ext-real-valued real-valued natural-valued Element of bool [:i,NAT:]
k1 is Element of i
{ H1(b1) where b1 is Element of i : ( b1 <=_ Fg,k1 & not b1 = k1 ) } is set
k2 is Element of i
k1 is Element of i
{ H1(b1) where b1 is Element of i : ( b1 <=_ Fg,k1 & not b1 = k1 ) } is set
k2 is set
i1 is Element of i
rng db is non empty finite countable V72() V73() V74() V77() Element of bool NAT
card i is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
Seg (card i) is non empty finite card i -element countable Element of bool NAT
k1 is set
dom db is non empty finite countable Element of bool i
bool i is non empty cup-closed diff-closed preBoolean finite V40() countable set
k2 is set
db . k2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
j1 is Element of i
{ H1(b1) where b1 is Element of i : S1[b1] } is set
{ H1(b1) where b1 is Element of i : ( b1 <=_ Fg,j1 & not b1 = j1 ) } is set
dbi1 is finite countable set
card dbi1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(card dbi1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(card dbi1) +^ 1 is epsilon-transitive epsilon-connected ordinal set
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
i1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
[:i,(Seg (card i)):] is Relation-like non empty finite countable set
bool [:i,(Seg (card i)):] is non empty cup-closed diff-closed preBoolean finite V40() countable set
M |_2 D is Relation-like set
[:D,D:] is Relation-like finite countable set
M /\ [:D,D:] is Relation-like k -defined k -valued finite countable set
(x,Fg,i) is Relation-like i -defined i -valued total quasi_total finite countable reflexive antisymmetric transitive Element of bool [:i,i:]
[:i,i:] is Relation-like non empty finite countable set
bool [:i,i:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
Fg /\ [:i,i:] is Relation-like x -defined x -valued finite countable set
field (x,Fg,i) is finite countable set
dom (x,Fg,i) is finite countable set
rng (x,Fg,i) is finite countable set
(dom (x,Fg,i)) \/ (rng (x,Fg,i)) is finite countable set
k2 is set
i1 is set
db . k2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
db . i1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
dbi1 is Element of i
{ H1(b1) where b1 is Element of i : S1[b1] } is set
{ H1(b1) where b1 is Element of i : ( b1 <=_ Fg,dbi1 & not b1 = dbi1 ) } is set
j1 is Element of i
{ H1(b1) where b1 is Element of i : S2[b1] } is set
{ H1(b1) where b1 is Element of i : ( b1 <=_ Fg,j1 & not b1 = j1 ) } is set
b12 is Element of i
b119 is Element of i
{ H1(b1) where b1 is Element of i : ( b1 <=_ Fg,b12 & not b1 = b12 ) } is set
{ H1(b1) where b1 is Element of i : ( b1 <=_ Fg,b119 & not b1 = b119 ) } is set
[b12,b119] is Element of [:i,i:]
{b12,b119} is non empty finite countable set
{b12} is non empty trivial finite 1 -element countable set
{{b12,b119},{b12}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
b129 is set
b111 is Element of D
[b111,b12] is set
{b111,b12} is non empty finite countable set
{b111} is non empty trivial finite 1 -element countable set
{{b111,b12},{b111}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
[b111,b119] is set
{b111,b119} is non empty finite countable set
{{b111,b119},{b111}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
db . dbi1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
ddbi11 is finite countable set
card ddbi11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(card ddbi11) +^ 1 is epsilon-transitive epsilon-connected ordinal set
(card ddbi11) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
b11 is finite countable set
card b11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(card b11) +^ 1 is epsilon-transitive epsilon-connected ordinal set
(card b11) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
[dbi1,j1] is Element of [:i,i:]
{dbi1,j1} is non empty finite countable set
{dbi1} is non empty trivial finite 1 -element countable set
{{dbi1,j1},{dbi1}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
[j1,dbi1] is Element of [:i,i:]
{j1,dbi1} is non empty finite countable set
{j1} is non empty trivial finite 1 -element countable set
{{j1,dbi1},{j1}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
b12 is Element of i
b119 is Element of i
{ H1(b1) where b1 is Element of i : ( b1 <=_ Fg,b119 & not b1 = b119 ) } is set
[b12,b119] is Element of [:i,i:]
{b12,b119} is non empty finite countable set
{b12} is non empty trivial finite 1 -element countable set
{{b12,b119},{b12}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
k1 is Relation-like i -defined Seg (card i) -valued Function-like non empty total quasi_total finite countable Element of bool [:i,(Seg (card i)):]
i1 is Element of i
db . i1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k2 is Element of i
db . k2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
{ H1(b1) where b1 is Element of i : S1[b1] } is set
{ H1(b1) where b1 is Element of i : ( b1 <=_ Fg,k2 & not b1 = k2 ) } is set
{ H1(b1) where b1 is Element of i : S2[b1] } is set
{ H1(b1) where b1 is Element of i : ( b1 <=_ Fg,i1 & not b1 = i1 ) } is set
j1 is finite countable set
card j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(card j1) +^ 1 is epsilon-transitive epsilon-connected ordinal set
(card j1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
dbi1 is finite countable set
card dbi1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(card dbi1) +^ 1 is epsilon-transitive epsilon-connected ordinal set
(card dbi1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
((card dbi1) + 1) - 1 is V55() real ext-real Element of REAL
((card j1) + 1) - 1 is V55() real ext-real Element of REAL
card (card dbi1) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
card (card j1) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
b11 is epsilon-transitive epsilon-connected ordinal cardinal set
ddbi11 is epsilon-transitive epsilon-connected ordinal cardinal set
dbi1 \ j1 is finite countable Element of bool dbi1
bool dbi1 is non empty cup-closed diff-closed preBoolean finite V40() countable set
b12 is set
b119 is Element of i
[b119,i1] is Element of [:i,i:]
{b119,i1} is non empty finite countable set
{b119} is non empty trivial finite 1 -element countable set
{{b119,i1},{b119}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
b129 is Element of i
[b119,k2] is Element of [:i,i:]
{b119,k2} is non empty finite countable set
{{b119,k2},{b119}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
b129 is Element of i
[k2,b119] is Element of [:i,i:]
{k2,b119} is non empty finite countable set
{k2} is non empty trivial finite 1 -element countable set
{{k2,b119},{k2}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
[k2,i1] is Element of [:i,i:]
{k2,i1} is non empty finite countable set
{{k2,i1},{k2}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
b129 is Element of i
card (Seg (card i)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
rng k1 is non empty finite countable Element of bool (Seg (card i))
bool (Seg (card i)) is non empty cup-closed diff-closed preBoolean finite V40() countable set
k1 " is Relation-like Function-like set
dom (k1 ") is set
dom k1 is non empty finite countable Element of bool i
bool i is non empty cup-closed diff-closed preBoolean finite V40() countable set
k2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
rng k2 is finite countable set
i1 is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
rng i1 is finite countable Element of bool k
dom i1 is finite countable V77() Element of bool NAT
j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
dbi1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
i1 /. j1 is Element of k
i1 /. dbi1 is Element of k
[(i1 /. j1),(i1 /. dbi1)] is set
{(i1 /. j1),(i1 /. dbi1)} is non empty finite countable set
{(i1 /. j1)} is non empty trivial finite 1 -element countable set
{{(i1 /. j1),(i1 /. dbi1)},{(i1 /. j1)}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
i1 . j1 is set
ddbi11 is Element of i
db . ddbi11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
i1 . dbi1 is set
b11 is Element of i
db . b11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
[ddbi11,b11] is Element of [:i,i:]
{ddbi11,b11} is non empty finite countable set
{ddbi11} is non empty trivial finite 1 -element countable set
{{ddbi11,b11},{ddbi11}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
x is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
rng x is finite countable Element of bool k
dom x is finite countable V77() Element of bool NAT
i is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
rng i is finite countable Element of bool k
dom i is finite countable V77() Element of bool NAT
Fg is non empty set
bool Fg is non empty cup-closed diff-closed preBoolean set
db is non empty finite countable Element of bool Fg
(db) is Relation-like non-empty empty-yielding NAT -defined db -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued Element of db *
db * is functional non empty FinSequence-membered FinSequenceSet of db
j1 is Relation-like NAT -defined db -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of db
dom j1 is finite countable V77() Element of bool NAT
rng j1 is finite countable Element of bool db
bool db is non empty cup-closed diff-closed preBoolean finite V40() countable set
dbi1 is Element of db
(db,dbi1) is Relation-like NAT -defined db -valued Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like Element of db *
j1 ^ (db,dbi1) is Relation-like NAT -defined Fg -valued Function-like non empty finite countable FinSequence-like FinSubsequence-like M26(Fg,db)
dom (j1 ^ (db,dbi1)) is non empty finite countable V77() Element of bool NAT
rng (j1 ^ (db,dbi1)) is non empty finite countable Element of bool db
ddbi11 is Relation-like NAT -defined db -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of db
rng ddbi11 is finite countable Element of bool db
dom ddbi11 is finite countable V77() Element of bool NAT
rng (j1 ^ (db,dbi1)) is non empty finite countable Element of bool Fg
len ddbi11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
b11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
b11 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
b12 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
b119 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len b119 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom b119 is finite countable V77() Element of bool NAT
b119 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len b119 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom b119 is finite countable V77() Element of bool NAT
dom (db,dbi1) is non empty trivial finite 1 -element countable V77() Element of bool NAT
b129 is Element of db
b111 is Element of db
[b111,b129] is Element of [:db,db:]
[:db,db:] is Relation-like non empty finite countable set
{b111,b129} is non empty finite countable set
{b111} is non empty trivial finite 1 -element countable set
{{b111,b129},{b111}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
b112 is set
(j1 ^ (db,dbi1)) . b112 is set
(j1 ^ (db,dbi1)) /. b112 is Element of db
i2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(len j1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(j1 ^ (db,dbi1)) . i2 is set
(db,dbi1) . 1 is set
len (j1 ^ (db,dbi1)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
Seg (len (j1 ^ (db,dbi1))) is non empty finite len (j1 ^ (db,dbi1)) -element countable Element of bool NAT
len (db,dbi1) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(len j1) + (len (db,dbi1)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
i2 - i2 is V55() real ext-real Element of REAL
1 - (len (j1 ^ (db,dbi1))) is V55() real ext-real Element of REAL
(j1 ^ (db,dbi1)) . ((len j1) + 1) is set
(j1 ^ (db,dbi1)) . ((len j1) + (len (db,dbi1))) is set
(j1 ^ (db,dbi1)) . (len (j1 ^ (db,dbi1))) is set
(j1 ^ (db,dbi1)) /. (len (j1 ^ (db,dbi1))) is Element of db
b129 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
(len b119) + b129 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
ddbi11 . ((len b119) + b129) is set
(db,dbi1) . b129 is set
{1} is non empty trivial finite V40() 1 -element countable with_non-empty_elements non empty-membered Element of bool NAT
b111 is Element of db
b111 is Element of db
Seg (len ddbi11) is finite len ddbi11 -element countable Element of bool NAT
ddbi11 . (len ddbi11) is set
i2 is Element of db
ddbi11 /. (len ddbi11) is Element of db
field M is set
dom M is set
rng M is set
(dom M) \/ (rng M) is set
{dbi1} is non empty trivial finite 1 -element countable Element of bool db
rng (db,dbi1) is non empty trivial finite 1 -element countable Element of bool db
(rng j1) \/ (rng (db,dbi1)) is non empty finite countable Element of bool db
j2 is set
ddbi11 . j2 is set
dbi2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
b112 is Element of db
[i2,b112] is Element of [:db,db:]
[:db,db:] is Relation-like non empty finite countable set
{i2,b112} is non empty finite countable set
{i2} is non empty trivial finite 1 -element countable set
{{i2,b112},{i2}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
ddbi11 /. dbi2 is Element of db
[b112,i2] is Element of [:db,db:]
{b112,i2} is non empty finite countable set
{b112} is non empty trivial finite 1 -element countable set
{{b112,i2},{b112}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
b12 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
b129 is set
rng b119 is finite countable set
b111 is set
b119 . b111 is set
Seg (len b119) is finite len b119 -element countable Element of bool NAT
b112 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
ddbi11 . b112 is set
Seg (b12 + 1) is non empty finite b12 + 1 -element b12 + 1 -element countable Element of bool NAT
b12 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
len (db,dbi1) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(len b119) + (len (db,dbi1)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
Seg ((len b119) + (len (db,dbi1))) is non empty finite (len b119) + (len (db,dbi1)) -element countable Element of bool NAT
b119 ^ (db,dbi1) is Relation-like NAT -defined Function-like non empty finite countable FinSequence-like FinSubsequence-like set
len j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(len j1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(len j1) + (len (db,dbi1)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
len (j1 ^ (db,dbi1)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
Seg (len (j1 ^ (db,dbi1))) is non empty finite len (j1 ^ (db,dbi1)) -element countable Element of bool NAT
b111 is set
j1 . b111 is set
Seg (len j1) is finite len j1 -element countable Element of bool NAT
b112 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(j1 ^ (db,dbi1)) . b112 is set
(j1 ^ (db,dbi1)) /. b112 is Element of db
(j1 ^ (db,dbi1)) . ((len j1) + 1) is set
(j1 ^ (db,dbi1)) /. ((len j1) + 1) is Element of db
{dbi1} is non empty trivial finite 1 -element countable Element of bool db
(rng j1) \/ {dbi1} is non empty finite countable Element of bool db
((rng j1) \/ {dbi1}) \ {dbi1} is finite countable Element of bool db
b111 is set
rng (db,dbi1) is non empty trivial finite 1 -element countable Element of bool db
(rng j1) \/ (rng (db,dbi1)) is non empty finite countable Element of bool db
(rng (j1 ^ (db,dbi1))) \ {dbi1} is finite countable Element of bool Fg
b129 is Relation-like NAT -defined db -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of db
rng b129 is finite countable Element of bool db
dom b129 is finite countable V77() Element of bool NAT
len b129 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(len b129) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(len b129) + (len (db,dbi1)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
b129 ^ (db,dbi1) is Relation-like NAT -defined Fg -valued Function-like non empty finite countable FinSequence-like FinSubsequence-like M26(Fg,db)
len (b129 ^ (db,dbi1)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
Seg (len (b129 ^ (db,dbi1))) is non empty finite len (b129 ^ (db,dbi1)) -element countable Element of bool NAT
dom (b129 ^ (db,dbi1)) is non empty finite countable V77() Element of bool NAT
b111 is set
b129 . b111 is set
Seg (len b129) is finite len b129 -element countable Element of bool NAT
b112 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
ddbi11 . b112 is set
ddbi11 /. b112 is Element of db
ddbi11 . ((len b129) + 1) is set
ddbi11 /. ((len b129) + 1) is Element of db
(rng b129) \/ {dbi1} is non empty finite countable Element of bool db
((rng b129) \/ {dbi1}) \ {dbi1} is finite countable Element of bool db
b111 is set
b129 ^ (db,dbi1) is Relation-like NAT -defined Fg -valued Function-like non empty finite countable FinSequence-like FinSubsequence-like M26(Fg,db)
rng (b129 ^ (db,dbi1)) is non empty finite countable Element of bool Fg
(rng b129) \/ (rng (db,dbi1)) is non empty finite countable Element of bool db
b111 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
b112 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
b129 /. b111 is Element of db
b129 /. b112 is Element of db
[(b129 /. b111),(b129 /. b112)] is Element of [:db,db:]
[:db,db:] is Relation-like non empty finite countable set
{(b129 /. b111),(b129 /. b112)} is non empty finite countable set
{(b129 /. b111)} is non empty trivial finite 1 -element countable set
{{(b129 /. b111),(b129 /. b112)},{(b129 /. b111)}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
b129 . b112 is set
ddbi11 . b112 is set
ddbi11 /. b112 is Element of db
b129 . b111 is set
ddbi11 . b111 is set
ddbi11 /. b111 is Element of db
b111 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
b112 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
j1 /. b111 is Element of db
j1 /. b112 is Element of db
[(j1 /. b111),(j1 /. b112)] is Element of [:db,db:]
[:db,db:] is Relation-like non empty finite countable set
{(j1 /. b111),(j1 /. b112)} is non empty finite countable set
{(j1 /. b111)} is non empty trivial finite 1 -element countable set
{{(j1 /. b111),(j1 /. b112)},{(j1 /. b111)}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
j1 . b112 is set
(j1 ^ (db,dbi1)) . b112 is set
(j1 ^ (db,dbi1)) /. b112 is Element of db
j1 . b111 is set
(j1 ^ (db,dbi1)) . b111 is set
(j1 ^ (db,dbi1)) /. b111 is Element of db
b111 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
b112 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
j1 /. b111 is Element of db
j1 /. b112 is Element of db
[(j1 /. b111),(j1 /. b112)] is Element of [:db,db:]
[:db,db:] is Relation-like non empty finite countable set
{(j1 /. b111),(j1 /. b112)} is non empty finite countable set
{(j1 /. b111)} is non empty trivial finite 1 -element countable set
{{(j1 /. b111),(j1 /. b112)},{(j1 /. b111)}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
k2 is Relation-like NAT -defined db -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of db
dom k2 is finite countable V77() Element of bool NAT
dbi1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
k2 /. dbi1 is Element of db
k2 . dbi1 is set
x /. dbi1 is Element of k
k2 /. j1 is Element of db
k2 . j1 is set
x /. j1 is Element of k
[(k2 /. j1),(k2 /. dbi1)] is Element of [:db,db:]
[:db,db:] is Relation-like non empty finite countable set
{(k2 /. j1),(k2 /. dbi1)} is non empty finite countable set
{(k2 /. j1)} is non empty trivial finite 1 -element countable set
{{(k2 /. j1),(k2 /. dbi1)},{(k2 /. j1)}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
i1 is Relation-like NAT -defined db -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of db
dom i1 is finite countable V77() Element of bool NAT
dbi1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
i1 /. dbi1 is Element of db
i1 . dbi1 is set
i /. dbi1 is Element of k
i1 /. j1 is Element of db
i1 . j1 is set
i /. j1 is Element of k
[(i1 /. j1),(i1 /. dbi1)] is Element of [:db,db:]
{(i1 /. j1),(i1 /. dbi1)} is non empty finite countable set
{(i1 /. j1)} is non empty trivial finite 1 -element countable set
{{(i1 /. j1),(i1 /. dbi1)},{(i1 /. j1)}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
dom (db) is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V77() Element of bool NAT
rng (db) is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty trivial proper finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued increasing decreasing non-decreasing non-increasing V72() V73() V74() V77() with_non-empty_elements Element of bool db
bool db is non empty cup-closed diff-closed preBoolean finite V40() countable set
j1 is Relation-like NAT -defined db -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of db
rng j1 is finite countable Element of bool db
dom j1 is finite countable V77() Element of bool NAT
k is set
bool k is non empty cup-closed diff-closed preBoolean set
[:k,k:] is Relation-like set
bool [:k,k:] is non empty cup-closed diff-closed preBoolean set
D is finite countable Element of bool k
M is Relation-like k -defined k -valued total quasi_total reflexive antisymmetric transitive Element of bool [:k,k:]
(k,D,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
x is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
rng x is finite countable Element of bool k
dom x is finite countable V77() Element of bool NAT
i is set
Fg is set
db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
x . db is set
k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
x . k1 is set
x /. k1 is Element of k
x /. db is Element of k
[i,Fg] is set
{i,Fg} is non empty finite countable set
{i} is non empty trivial finite 1 -element countable set
{{i,Fg},{i}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
[Fg,i] is set
{Fg,i} is non empty finite countable set
{Fg} is non empty trivial finite 1 -element countable set
{{Fg,i},{Fg}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
field M is set
dom M is set
rng M is set
(dom M) \/ (rng M) is set
k is set
Fin k is non empty cup-closed diff-closed preBoolean set
bool (Fin k) is non empty cup-closed diff-closed preBoolean set
D is non empty Element of bool (Fin k)
M is Element of D
k is set
Fin k is non empty cup-closed diff-closed preBoolean set
bool (Fin k) is non empty cup-closed diff-closed preBoolean set
D is non empty Element of bool (Fin k)
bool k is non empty cup-closed diff-closed preBoolean set
M is finite countable Element of D
k is set
bool k is non empty cup-closed diff-closed preBoolean set
[:k,k:] is Relation-like set
bool [:k,k:] is non empty cup-closed diff-closed preBoolean set
D is finite countable Element of bool k
M is Relation-like k -defined k -valued total quasi_total reflexive antisymmetric transitive Element of bool [:k,k:]
(k,D,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
rng (k,D,M) is finite countable Element of bool k
i is Relation-like NAT -defined D -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of D
Fg is set
dom i is finite countable V77() set
db is set
i . Fg is set
i . db is set
dom i is finite countable V77() Element of bool NAT
k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
i . k1 is set
k2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
i . k2 is set
j1 is Element of D
i /. k2 is Element of D
i1 is Element of D
i /. k1 is Element of D
(k,D,M) /. k2 is Element of k
(k,D,M) /. k1 is Element of k
k is set
bool k is non empty cup-closed diff-closed preBoolean set
[:k,k:] is Relation-like set
bool [:k,k:] is non empty cup-closed diff-closed preBoolean set
D is finite countable Element of bool k
card D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M is Relation-like k -defined k -valued total quasi_total reflexive antisymmetric transitive Element of bool [:k,k:]
(k,D,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
len (k,D,M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom (k,D,M) is finite countable V77() Element of bool NAT
Seg (len (k,D,M)) is finite len (k,D,M) -element countable Element of bool NAT
card (Seg (len (k,D,M))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
card (len (k,D,M)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
rng (k,D,M) is finite countable Element of bool k
k is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
k + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
D is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Del (D,(k + 1)) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len (Del (D,(k + 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Seg (len D) is finite len D -element countable Element of bool NAT
dom D is finite countable V77() Element of bool NAT
k is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
k + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
D is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Del (D,(len D)) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
D . (len D) is set
<*(D . (len D))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like set
(Del (D,(len D))) ^ <*(D . (len D))*> is Relation-like NAT -defined Function-like non empty finite countable FinSequence-like FinSubsequence-like set
M is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
x is set
<*x*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like set
M ^ <*x*> is Relation-like NAT -defined Function-like non empty finite countable FinSequence-like FinSubsequence-like set
len M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len <*x*> is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(len M) + (len <*x*>) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(len M) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
len (Del (D,(len D))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom M is finite countable V77() Element of bool NAT
Seg (len M) is finite len M -element countable Element of bool NAT
i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
(Del (D,(len D))) . i is set
D . i is set
M . i is set
the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
the set is set
{ the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set } is functional non empty trivial finite V40() 1 -element with_common_domain countable set
[: the set ,{ the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set }:] is Relation-like set
bool [: the set ,{ the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set }:] is non empty cup-closed diff-closed preBoolean set
the set --> the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set is Relation-like the set -defined { the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set } -valued Function-like constant total quasi_total Function-yielding V50() Element of bool [: the set ,{ the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set }:]
M is Relation-like the set -defined { the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set } -valued Function-like constant total quasi_total Function-yielding V50() Element of bool [: the set ,{ the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set }:]
x is set
dom M is set
M . x is Relation-like Function-like set
dom M is Element of bool the set
bool the set is non empty cup-closed diff-closed preBoolean set
k is Relation-like Function-like () set
dom k is set
D is Relation-like Function-like () set
dom D is set
(dom k) /\ (dom D) is set
M is set
k . M is set
D . M is set
x is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
i is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
x ^ i is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
Fg is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
db is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
Fg ^ db is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
M is Relation-like Function-like set
dom M is set
x is set
M . x is set
k . x is set
D . x is set
i is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
Fg is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
i ^ Fg is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
x is Relation-like Function-like () set
dom x is set
i is set
Fg is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
k . i is set
db is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
D . i is set
x . i is set
Fg ^ db is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
M is Relation-like Function-like () set
dom M is set
x is Relation-like Function-like () set
dom x is set
i is set
k . i is set
D . i is set
M . i is set
Fg is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
db is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
Fg ^ db is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
x . i is set
k is non empty set
D is non empty set
PFuncs (k,D) is functional non empty set
M is set
x is set
[M,x] is set
{M,x} is non empty finite countable set
{M} is non empty trivial finite 1 -element countable set
{{M,x},{M}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
{[M,x]} is Relation-like Function-like constant non empty trivial finite 1 -element countable set
dom {[M,x]} is non empty trivial finite 1 -element countable set
{x} is non empty trivial finite 1 -element countable set
rng {[M,x]} is non empty trivial finite 1 -element countable set
Fg is Relation-like Function-like Element of PFuncs (k,D)
k is set
D is set
PFuncs (k,D) is functional non empty set
M is Relation-like Function-like Element of PFuncs (k,D)
x is set
i is Relation-like Function-like set
dom i is set
rng i is set
Fg is Relation-like Function-like set
rng Fg is set
dom Fg is set
k is set
D is set
PFuncs (k,D) is functional non empty set
[:k,D:] is Relation-like set
bool [:k,D:] is non empty cup-closed diff-closed preBoolean Element of bool (bool [:k,D:])
bool [:k,D:] is non empty cup-closed diff-closed preBoolean set
bool (bool [:k,D:]) is non empty cup-closed diff-closed preBoolean set
M is set
x is Relation-like Function-like set
dom x is set
rng x is set
[:(dom x),(rng x):] is Relation-like set
k is set
D is set
PFuncs (k,D) is functional non empty set
[:k,D:] is Relation-like set
bool [:k,D:] is non empty cup-closed diff-closed preBoolean Element of bool (bool [:k,D:])
bool [:k,D:] is non empty cup-closed diff-closed preBoolean set
bool (bool [:k,D:]) is non empty cup-closed diff-closed preBoolean set
the non empty finite countable set is non empty finite countable set
PFuncs ( the non empty finite countable set , the non empty finite countable set ) is functional non empty set
D is functional non empty set
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence of k *
M is set
dom D is finite countable V77() set
D . M is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom D is finite countable V77() Element of bool NAT
k is Relation-like Function-like set
D is set
dom k is set
k . D is set
k is set
[:k,k:] is Relation-like set
bool [:k,k:] is non empty cup-closed diff-closed preBoolean set
D is Relation-like set
field D is set
dom D is set
rng D is set
(dom D) \/ (rng D) is set
M is set
x is set
[M,x] is set
{M,x} is non empty finite countable set
{M} is non empty trivial finite 1 -element countable set
{{M,x},{M}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
k is set
D is Relation-like k -defined Function-like total set
M is set
x is set
D +* (M,x) is Relation-like Function-like set
dom (D +* (M,x)) is set
dom D is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
k is set
D is Relation-like k -defined Function-like total set
M is set
x is set
D +* (M,x) is Relation-like k -defined Function-like set
dom (D +* (M,x)) is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
dom D is Element of bool k
i is Relation-like k -defined Function-like set
k is Relation-like Function-like one-to-one set
card k is epsilon-transitive epsilon-connected ordinal cardinal set
rng k is set
card (rng k) is epsilon-transitive epsilon-connected ordinal cardinal set
k " is Relation-like Function-like one-to-one set
dom (k ") is set
dom k is set
rng (k ") is set
card (dom k) is epsilon-transitive epsilon-connected ordinal cardinal set
card (rng (k ")) is epsilon-transitive epsilon-connected ordinal cardinal set
card (dom (k ")) is epsilon-transitive epsilon-connected ordinal cardinal set
k is set
D is set
M is functional non empty FinSequence-membered FinSequenceSet of k
[:D,M:] is Relation-like set
bool [:D,M:] is non empty cup-closed diff-closed preBoolean set
x is Relation-like D -defined M -valued Function-like Function-yielding V50() Element of bool [:D,M:]
i is set
x /. i is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like Element of M
k is set
[:k,k:] is Relation-like set
bool [:k,k:] is non empty cup-closed diff-closed preBoolean set
D is Relation-like set
field D is set
dom D is set
rng D is set
(dom D) \/ (rng D) is set
M is Relation-like k -defined k -valued Element of bool [:k,k:]
dom M is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
x is Relation-like k -defined k -valued total quasi_total reflexive antisymmetric transitive Element of bool [:k,k:]
k is non empty set
bool k is non empty cup-closed diff-closed preBoolean set
[:k,k:] is Relation-like non empty set
bool [:k,k:] is non empty cup-closed diff-closed preBoolean set
D is non empty finite countable Element of bool k
M is Relation-like k -defined k -valued total quasi_total reflexive antisymmetric transitive Element of bool [:k,k:]
(k,D,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
(k,D,M) /. 1 is Element of k
x is Element of k
rng (k,D,M) is finite countable Element of bool k
dom (k,D,M) is finite countable V77() Element of bool NAT
i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,D,M) /. i is Element of k
[x,((k,D,M) /. 1)] is Element of [:k,k:]
{x,((k,D,M) /. 1)} is non empty finite countable set
{x} is non empty trivial finite 1 -element countable set
{{x,((k,D,M) /. 1)},{x}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
field M is set
dom M is set
rng M is set
(dom M) \/ (rng M) is set
[((k,D,M) /. 1),x] is Element of [:k,k:]
{((k,D,M) /. 1),x} is non empty finite countable set
{((k,D,M) /. 1)} is non empty trivial finite 1 -element countable set
{{((k,D,M) /. 1),x},{((k,D,M) /. 1)}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
k is non empty set
bool k is non empty cup-closed diff-closed preBoolean set
[:k,k:] is Relation-like non empty set
bool [:k,k:] is non empty cup-closed diff-closed preBoolean set
D is non empty finite countable Element of bool k
M is Relation-like k -defined k -valued total quasi_total reflexive antisymmetric transitive Element of bool [:k,k:]
(k,D,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
len (k,D,M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,D,M) /. (len (k,D,M)) is Element of k
x is Element of k
rng (k,D,M) is finite countable Element of bool k
dom (k,D,M) is finite countable V77() Element of bool NAT
Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,D,M) /. Fg is Element of k
[((k,D,M) /. (len (k,D,M))),x] is Element of [:k,k:]
{((k,D,M) /. (len (k,D,M))),x} is non empty finite countable set
{((k,D,M) /. (len (k,D,M)))} is non empty trivial finite 1 -element countable set
{{((k,D,M) /. (len (k,D,M))),x},{((k,D,M) /. (len (k,D,M)))}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
field M is set
dom M is set
rng M is set
(dom M) \/ (rng M) is set
[x,((k,D,M) /. (len (k,D,M)))] is Element of [:k,k:]
{x,((k,D,M) /. (len (k,D,M)))} is non empty finite countable set
{x} is non empty trivial finite 1 -element countable set
{{x,((k,D,M) /. (len (k,D,M)))},{x}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
k is non empty set
bool k is non empty cup-closed diff-closed preBoolean set
[:k,k:] is Relation-like non empty set
bool [:k,k:] is non empty cup-closed diff-closed preBoolean set
D is non empty finite countable Element of bool k
M is Relation-like k -defined k -valued total quasi_total reflexive antisymmetric transitive being_linear-order Element of bool [:k,k:]
(k,D,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
field M is set
dom M is set
rng M is set
(dom M) \/ (rng M) is set
k is Relation-like Function-like set
D is set
dom k is set
k . D is set
k is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () set
D is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () set
(k,D) is Relation-like Function-like Function-yielding V50() () set
dom (k,D) is set
dom k is finite countable V77() Element of bool NAT
dom D is finite countable V77() Element of bool NAT
(dom k) /\ (dom D) is finite countable Element of bool NAT
len k is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Seg (len k) is finite len k -element countable Element of bool NAT
(Seg (len k)) /\ (dom D) is finite countable Element of bool NAT
len D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Seg (len D) is finite len D -element countable Element of bool NAT
(Seg (len k)) /\ (Seg (len D)) is finite countable Element of bool NAT
min ((len k),(len D)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Seg (min ((len k),(len D))) is finite min ((len k),(len D)) -element countable Element of bool NAT
k is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
D is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
k |-> D is Relation-like NAT -defined Function-like finite k -element countable FinSequence-like FinSubsequence-like set
Seg k is finite k -element countable Element of bool NAT
(Seg k) --> D is Relation-like Seg k -defined Seg k -defined {D} -valued Function-like constant total total quasi_total finite countable Function-yielding V50() FinSequence-like FinSubsequence-like Element of bool [:(Seg k),{D}:]
{D} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
[:(Seg k),{D}:] is Relation-like finite countable set
bool [:(Seg k),{D}:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
M is set
dom (k |-> D) is finite k -element countable V77() set
(k |-> D) . M is set
dom (k |-> D) is finite k -element countable V77() Element of bool NAT
k is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () set
D is set
k . D is Relation-like Function-like set
dom k is finite countable V77() Element of bool NAT
dom k is finite countable V77() Element of bool NAT
dom k is finite countable V77() Element of bool NAT
k is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
Card k is Relation-like Function-like Cardinal-yielding set
dom (Card k) is set
dom k is finite countable V77() Element of bool NAT
len k is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Seg (len k) is finite len k -element countable Element of bool NAT
k is Relation-like Function-like set
rng k is set
D is set
dom k is set
M is set
k . M is set
D is set
k . D is set
k is Relation-like NAT -defined Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like set
D is Relation-like NAT -defined Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like set
k ^ D is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
rng (k ^ D) is finite countable set
rng k is finite countable set
rng D is finite countable set
(rng k) \/ (rng D) is finite countable set
M is set
k is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
D is set
dom k is finite countable V77() set
k . D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
dom k is finite countable V77() Element of bool NAT
(NAT) is Relation-like non-empty empty-yielding NAT -defined NAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () Element of NAT *
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
Card D is Relation-like NAT -defined Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like set
rng (Card D) is finite countable set
M is set
dom (Card D) is finite countable V77() Element of bool NAT
x is set
(Card D) . x is set
D . x is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom D is finite countable V77() Element of bool NAT
i is finite countable set
card i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
k | D is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
k is Relation-like Function-like set
Card k is Relation-like Function-like Cardinal-yielding set
D is set
k | D is Relation-like Function-like set
Card (k | D) is Relation-like Function-like Cardinal-yielding set
(Card k) | D is Relation-like Function-like Cardinal-yielding set
dom ((Card k) | D) is set
dom (Card k) is set
(dom (Card k)) /\ D is set
dom k is set
(dom k) /\ D is set
dom (k | D) is set
M is set
((Card k) | D) . M is set
(Card k) . M is set
k . M is set
card (k . M) is epsilon-transitive epsilon-connected ordinal cardinal set
(k | D) . M is set
card ((k | D) . M) is epsilon-transitive epsilon-connected ordinal cardinal set
k is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () set
Card k is Relation-like NAT -defined Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like set
dom k is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V77() () Element of bool NAT
dom (Card k) is finite countable V77() Element of bool NAT
k is set
<*k*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like set
Card <*k*> is Relation-like NAT -defined Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like set
card k is epsilon-transitive epsilon-connected ordinal cardinal set
<*(card k)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like set
dom <*(card k)*> is non empty trivial finite 1 -element countable V77() Element of bool NAT
{1} is non empty trivial finite V40() 1 -element countable with_non-empty_elements non empty-membered Element of bool NAT
M is set
<*(card k)*> . M is set
dom <*k*> is non empty trivial finite 1 -element countable V77() Element of bool NAT
x is set
<*(card k)*> . x is set
<*k*> . x is set
card (<*k*> . x) is epsilon-transitive epsilon-connected ordinal cardinal set
M is Relation-like Function-like Cardinal-yielding set
k is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
D is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
k ^ D is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
Card (k ^ D) is Relation-like NAT -defined Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like set
Card k is Relation-like NAT -defined Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like set
Card D is Relation-like NAT -defined Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like set
(Card k) ^ (Card D) is Relation-like NAT -defined Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like set
dom (Card D) is finite countable V77() Element of bool NAT
dom D is finite countable V77() Element of bool NAT
len (Card D) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom (Card k) is finite countable V77() Element of bool NAT
dom k is finite countable V77() Element of bool NAT
len (Card k) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len k is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M is set
dom (k ^ D) is finite countable V77() Element of bool NAT
(len k) + (len D) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Seg ((len k) + (len D)) is finite (len k) + (len D) -element countable Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((Card k) ^ (Card D)) . M is set
(Card k) . x is set
k . x is set
card (k . x) is epsilon-transitive epsilon-connected ordinal cardinal set
(k ^ D) . M is set
card ((k ^ D) . M) is epsilon-transitive epsilon-connected ordinal cardinal set
i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
(len k) + i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((Card k) ^ (Card D)) . M is set
(Card D) . i is set
D . i is set
card (D . i) is epsilon-transitive epsilon-connected ordinal cardinal set
(k ^ D) . M is set
card ((k ^ D) . M) is epsilon-transitive epsilon-connected ordinal cardinal set
dom ((Card k) ^ (Card D)) is finite countable V77() Element of bool NAT
(len (Card k)) + (len (Card D)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Seg ((len (Card k)) + (len (Card D))) is finite (len (Card k)) + (len (Card D)) -element countable Element of bool NAT
k is set
(k) is Relation-like non-empty empty-yielding NAT -defined k -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () Element of k *
k * is functional non empty FinSequence-membered FinSequenceSet of k
k is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
<*k*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like set
D is set
dom <*k*> is non empty trivial finite 1 -element countable V77() set
<*k*> . D is set
dom <*k*> is non empty trivial finite 1 -element countable V77() Element of bool NAT
{1} is non empty trivial finite V40() 1 -element countable with_non-empty_elements non empty-membered Element of bool NAT
k is Relation-like Function-like set
rng k is set
D is set
dom k is set
M is set
k . M is set
D is set
k . D is set
k is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () set
D is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () set
k ^ D is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
rng (k ^ D) is finite countable set
rng k is finite countable set
rng D is finite countable set
(rng k) \/ (rng D) is finite countable set
M is set
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is Relation-like non-empty empty-yielding NAT -defined k * -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () FinSequence of k *
[:NAT,(k *):] is Relation-like non empty non trivial non finite non empty-membered set
(k,D) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
((k *)) is Relation-like non-empty empty-yielding NAT -defined k * -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () Element of (k *) *
(k *) * is functional non empty FinSequence-membered FinSequenceSet of k *
(k) is Relation-like non-empty empty-yielding NAT -defined k -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () Element of k *
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(k,D) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
len (k,D) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,D) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum (k,D) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
((k *),M) is Relation-like NAT -defined k * -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k *) *
(k *) * is functional non empty FinSequence-membered FinSequenceSet of k *
D ^ ((k *),M) is Relation-like NAT -defined k * -valued Function-like non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(k,(D ^ ((k *),M))) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
len (k,(D ^ ((k *),M))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,((k *),M)) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,(k,D),(k,((k *),M))) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
len (k,(k,D),(k,((k *),M))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len (k,((k *),M)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum (k,D)) + (len (k,((k *),M))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum (k,D)) + (len M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(NAT,(len M)) is Relation-like NAT -defined NAT -valued Function-like one-to-one constant non empty trivial finite 1 -element Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued increasing decreasing non-decreasing non-increasing Element of NAT *
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
(k,D) ^ (NAT,(len M)) is Relation-like NAT -defined NAT -valued Function-like non empty finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum ((k,D) ^ (NAT,(len M))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,((k *),M)) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(k,D) ^ (k,((k *),M)) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum ((k,D) ^ (k,((k *),M))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,(D ^ ((k *),M))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum (k,(D ^ ((k *),M))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((k *)) is Relation-like non-empty empty-yielding NAT -defined k * -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () Element of (k *) *
(k,((k *))) is Relation-like non-empty empty-yielding NAT -defined k -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () Element of k *
len (k,((k *))) is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () Element of NAT
(k,((k *))) is Relation-like non-empty empty-yielding NAT -defined NAT -valued RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () FinSequence of NAT
Sum (k,((k *))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is set
D * is functional non empty FinSequence-membered FinSequenceSet of D
M is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(k,M) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(k,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
len (k,M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x is Relation-like NAT -defined D * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of D *
(D,x) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(D,x) is Relation-like NAT -defined D -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of D *
len (D,x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Sum (D,x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
((k *)) is Relation-like non-empty empty-yielding NAT -defined k * -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () Element of (k *) *
(k *) * is functional non empty FinSequence-membered FinSequenceSet of k *
M is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(k,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
dom (k,M) is finite countable V77() Element of bool NAT
dom M is finite countable V77() Element of bool NAT
x is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
((k *),x) is Relation-like NAT -defined k * -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k *) *
M ^ ((k *),x) is Relation-like NAT -defined k * -valued Function-like non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(k,(M ^ ((k *),x))) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
dom (k,(M ^ ((k *),x))) is finite countable V77() Element of bool NAT
dom (M ^ ((k *),x)) is non empty finite countable V77() Element of bool NAT
i is set
(k,(M ^ ((k *),x))) . i is set
(k,((k *),x)) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,(k,M),(k,((k *),x))) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,(k,M),x) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,M) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum (k,M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len (k,M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(M ^ ((k *),x)) | (len M) is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(len M) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
db is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(M ^ ((k *),x)) . db is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom ((M ^ ((k *),x)) . db) is finite countable V77() Element of bool NAT
db -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(M ^ ((k *),x)) | (db -' 1) is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(k,((M ^ ((k *),x)) | (db -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum (k,((M ^ ((k *),x)) | (db -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Fg -' (Sum (k,((M ^ ((k *),x)) | (db -' 1)))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum (k,((M ^ ((k *),x)) | (db -' 1)))) + k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((M ^ ((k *),x)) . db) . k1 is set
len (M ^ ((k *),x)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
len ((k *),x) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(len M) + (len ((k *),x)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
len (k,(k,M),x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(len (k,M)) + (len x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom ((k *),x) is non empty trivial finite 1 -element countable V77() Element of bool NAT
((k *),x) . 1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(len (k,M)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(k,M) . i is set
db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M . db is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom (M . db) is finite countable V77() Element of bool NAT
db -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M | (db -' 1) is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(k,(M | (db -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum (k,(M | (db -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum (k,(M | (db -' 1)))) + k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(M . db) . k1 is set
(M ^ ((k *),x)) . db is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom ((M ^ ((k *),x)) . db) is finite countable V77() Element of bool NAT
(M ^ ((k *),x)) | (db -' 1) is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(k,((M ^ ((k *),x)) | (db -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum (k,((M ^ ((k *),x)) | (db -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum (k,((M ^ ((k *),x)) | (db -' 1)))) + k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((M ^ ((k *),x)) . db) . k1 is set
(k,((k *))) is Relation-like non-empty empty-yielding NAT -defined k -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () Element of k *
dom (k,((k *))) is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V77() () Element of bool NAT
dom ((k *)) is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V77() () Element of bool NAT
M is set
(k,((k *))) . M is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () set
k is set
k * is functional non empty FinSequence-membered FinSequenceSet of k
((k *)) is Relation-like non-empty empty-yielding NAT -defined k * -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () Element of (k *) *
(k *) * is functional non empty FinSequence-membered FinSequenceSet of k *
M is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
dom M is finite countable V77() Element of bool NAT
(k,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
dom (k,M) is finite countable V77() Element of bool NAT
x is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
((k *),x) is Relation-like NAT -defined k * -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k *) *
M ^ ((k *),x) is Relation-like NAT -defined k * -valued Function-like non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
dom (M ^ ((k *),x)) is non empty finite countable V77() Element of bool NAT
(k,(M ^ ((k *),x))) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
dom (k,(M ^ ((k *),x))) is finite countable V77() Element of bool NAT
i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(M ^ ((k *),x)) . i is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom ((M ^ ((k *),x)) . i) is finite countable V77() Element of bool NAT
i -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(M ^ ((k *),x)) | (i -' 1) is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(k,((M ^ ((k *),x)) | (i -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum (k,((M ^ ((k *),x)) | (i -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum (k,((M ^ ((k *),x)) | (i -' 1)))) + Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((M ^ ((k *),x)) . i) . Fg is set
(k,(M ^ ((k *),x))) . ((Sum (k,((M ^ ((k *),x)) | (i -' 1)))) + Fg) is set
(k,((k *),x)) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,(k,M),(k,((k *),x))) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
(k,(k,M),x) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
len M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(len M) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
len (M ^ ((k *),x)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
len ((k *),x) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(len M) + (len ((k *),x)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(k,M) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum (k,M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len (k,M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom ((k *),x) is non empty trivial finite 1 -element countable V77() Element of bool NAT
((k *),x) . 1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
M . i is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom (M . i) is finite countable V77() Element of bool NAT
M | (i -' 1) is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(k,(M | (i -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum (k,(M | (i -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum (k,(M | (i -' 1)))) + Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(M . i) . Fg is set
(k,M) . ((Sum (k,(M | (i -' 1)))) + Fg) is set
dom ((k *)) is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V77() () Element of bool NAT
(k,((k *))) is Relation-like non-empty empty-yielding NAT -defined k -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () Element of k *
dom (k,((k *))) is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V77() () Element of bool NAT
M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((k *)) . M is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () set
dom (((k *)) . M) is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V77() () Element of bool NAT
M -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((k *)) | (M -' 1) is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(k,(((k *)) | (M -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum (k,(((k *)) | (M -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum (k,(((k *)) | (M -' 1)))) + x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(((k *)) . M) . x is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative V61() complex-valued ext-real-valued real-valued natural-valued () set
(k,((k *))) . ((Sum (k,(((k *)) | (M -' 1)))) + x) is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () set
k is non empty set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is non empty set
[:k,D:] is Relation-like non empty set
bool [:k,D:] is non empty cup-closed diff-closed preBoolean set
D * is functional non empty FinSequence-membered FinSequenceSet of D
M is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
dom M is finite countable V77() Element of bool NAT
x is Relation-like k -defined D -valued Function-like non empty total quasi_total Element of bool [:k,D:]
(dom M) --> x is Relation-like non-empty dom M -defined bool [:k,D:] -valued Function-like constant total quasi_total finite countable Function-yielding V50() Element of bool [:(dom M),(bool [:k,D:]):]
[:(dom M),(bool [:k,D:]):] is Relation-like set
bool [:(dom M),(bool [:k,D:]):] is non empty cup-closed diff-closed preBoolean set
{x} is functional non empty trivial finite 1 -element with_common_domain countable with_non-empty_elements non empty-membered set
[:(dom M),{x}:] is Relation-like finite countable set
((dom M) --> x) ** M is Relation-like Function-like set
dom (((dom M) --> x) ** M) is set
dom ((dom M) --> x) is finite countable Element of bool (dom M)
bool (dom M) is non empty cup-closed diff-closed preBoolean finite V40() countable set
(dom ((dom M) --> x)) /\ (dom M) is finite countable Element of bool NAT
(dom M) /\ (dom M) is finite countable Element of bool NAT
rng (((dom M) --> x) ** M) is set
Fg is set
db is set
(((dom M) --> x) ** M) . db is set
k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M . k1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
((dom M) --> x) . k1 is Relation-like Function-like set
(M . k1) * (((dom M) --> x) . k1) is Relation-like NAT -defined Function-like finite countable set
(M . k1) * x is Relation-like NAT -defined D -valued Function-like finite countable set
len M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
Seg (len M) is finite len M -element countable Element of bool NAT
k is non empty set
k * is functional non empty FinSequence-membered FinSequenceSet of k
D is non empty set
[:k,D:] is Relation-like non empty set
bool [:k,D:] is non empty cup-closed diff-closed preBoolean set
D * is functional non empty FinSequence-membered FinSequenceSet of D
M is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
dom M is finite countable V77() Element of bool NAT
(k,M) is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k *
x is Relation-like k -defined D -valued Function-like non empty total quasi_total Element of bool [:k,D:]
(dom M) --> x is Relation-like non-empty dom M -defined bool [:k,D:] -valued Function-like constant total quasi_total finite countable Function-yielding V50() Element of bool [:(dom M),(bool [:k,D:]):]
[:(dom M),(bool [:k,D:]):] is Relation-like set
bool [:(dom M),(bool [:k,D:]):] is non empty cup-closed diff-closed preBoolean set
{x} is functional non empty trivial finite 1 -element with_common_domain countable with_non-empty_elements non empty-membered set
[:(dom M),{x}:] is Relation-like finite countable set
((dom M) --> x) ** M is Relation-like Function-like set
x * (k,M) is Relation-like NAT -defined D -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of D
i is Relation-like NAT -defined D * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of D *
(D,i) is Relation-like NAT -defined D -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of D *
len (k,M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k1 is Relation-like NAT -defined D -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of D
len k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom (k,M) is finite countable V77() Element of bool NAT
dom k1 is finite countable V77() Element of bool NAT
dom i is finite countable V77() Element of bool NAT
dom ((dom M) --> x) is finite countable Element of bool (dom M)
bool (dom M) is non empty cup-closed diff-closed preBoolean finite V40() countable set
(dom ((dom M) --> x)) /\ (dom M) is finite countable Element of bool NAT
(dom M) /\ (dom M) is finite countable Element of bool NAT
k2 is set
i . k2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
M . k2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
((dom M) --> x) . k2 is Relation-like Function-like set
(M . k2) * (((dom M) --> x) . k2) is Relation-like NAT -defined Function-like finite countable set
(M . k2) * x is Relation-like NAT -defined D -valued Function-like finite countable set
i1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M . i1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(k,M) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(k,M) . k2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
j1 is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
len j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len (i . k2) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(D,i) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(D,i) . k2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
dom (k,M) is finite countable V77() Element of bool NAT
dom (D,i) is finite countable V77() Element of bool NAT
len (D,i) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom (D,i) is finite countable V77() Element of bool NAT
k2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
(k,M) . k2 is set
i1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M . i1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom (M . i1) is finite countable V77() Element of bool NAT
i1 -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M | (i1 -' 1) is Relation-like NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of k *
(k,(M | (i1 -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Sum (k,(M | (i1 -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum (k,(M | (i1 -' 1)))) + j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(M . i1) . j1 is set
i . i1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
((dom M) --> x) . i1 is Relation-like Function-like set
(M . i1) * (((dom M) --> x) . i1) is Relation-like NAT -defined Function-like finite countable set
(M . i1) * x is Relation-like NAT -defined D -valued Function-like finite countable set
dbi1 is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of k
len dbi1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
len (i . i1) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom (i . i1) is finite countable V77() Element of bool NAT
dom x is non empty Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
rng (M . i1) is finite countable set
dom ((M . i1) * x) is finite countable V77() Element of bool NAT
i | (i1 -' 1) is Relation-like NAT -defined D * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of D *
(D,(i | (i1 -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Seg (i1 -' 1) is finite i1 -' 1 -element countable Element of bool NAT
i | (Seg (i1 -' 1)) is Relation-like NAT -defined Seg (i1 -' 1) -defined NAT -defined D * -valued Function-like finite countable Function-yielding V50() FinSubsequence-like Element of bool [:NAT,(D *):]
[:NAT,(D *):] is Relation-like non empty non trivial non finite non empty-membered set
bool [:NAT,(D *):] is non empty non trivial cup-closed diff-closed preBoolean non finite non empty-membered set
Card (i | (Seg (i1 -' 1))) is Relation-like Function-like Cardinal-yielding set
(k,M) | (Seg (i1 -' 1)) is Relation-like NAT -defined Seg (i1 -' 1) -defined NAT -defined NAT -valued RAT -valued Function-like finite Cardinal-yielding countable FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:NAT,NAT:]
bool [:NAT,NAT:] is non empty non trivial cup-closed diff-closed preBoolean non finite non empty-membered set
M | (Seg (i1 -' 1)) is Relation-like NAT -defined Seg (i1 -' 1) -defined NAT -defined k * -valued Function-like finite countable Function-yielding V50() FinSubsequence-like Element of bool [:NAT,(k *):]
[:NAT,(k *):] is Relation-like non empty non trivial non finite non empty-membered set
bool [:NAT,(k *):] is non empty non trivial cup-closed diff-closed preBoolean non finite non empty-membered set
Card (M | (Seg (i1 -' 1))) is Relation-like Function-like Cardinal-yielding set
(D,i) . k2 is set
(i . i1) . j1 is set
((M . i1) * (((dom M) --> x) . i1)) . j1 is set
((M . i1) * x) . j1 is set
x . ((M . i1) . j1) is set
k1 . k2 is set
k is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
D is set
M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
k +* (D,M) is Relation-like Function-like set
i is set
dom (k +* (D,M)) is set
(k +* (D,M)) . i is set
dom k is set
dom k is set
k . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
dom k is set
k . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
dom k is set
k is Relation-like Function-like complex-valued ext-real-valued real-valued set
D is set
M is V55() real ext-real set
k +* (D,M) is Relation-like Function-like set
i is set
dom (k +* (D,M)) is set
(k +* (D,M)) . i is set
dom k is set
dom k is set
k . i is V55() real ext-real set
dom k is set
k . i is V55() real ext-real set
dom k is set
k is set
D is Relation-like k -defined Function-like total complex-valued set
M is Relation-like k -defined Function-like total complex-valued set
D + M is Relation-like k -defined Function-like total complex-valued set
x is Relation-like k -defined Function-like total set
dom D is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
dom M is Element of bool k
dom x is Element of bool k
(dom D) /\ (dom M) is Element of bool k
i is set
x . i is set
D . i is V55() set
M . i is V55() set
(D . i) + (M . i) is Element of COMPLEX
i is set
x . i is set
D . i is V55() set
M . i is V55() set
(D . i) + (M . i) is Element of COMPLEX
k is set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
x is Relation-like k -defined Function-like total set
i is set
x . i is set
D . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(D . i) -' (M . i) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom M is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
dom D is Element of bool k
dom x is Element of bool k
0 -' 0 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
0 -' (M . i) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x is Relation-like k -defined Function-like total set
i is Relation-like k -defined Function-like total set
Fg is set
x . Fg is set
D . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(D . Fg) -' (M . Fg) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
i . Fg is set
k is set
D is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
M is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
x is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
(k,M,x) is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
i is set
D . i is V55() real ext-real set
M . i is V55() real ext-real set
x . i is V55() real ext-real set
(M . i) + (x . i) is V55() real ext-real Element of REAL
i is set
dom x is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
dom M is Element of bool k
dom D is Element of bool k
D . i is V55() real ext-real set
0 + 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () Element of NAT
x . i is V55() real ext-real set
0 + (x . i) is V55() real ext-real Element of REAL
M . i is V55() real ext-real set
(M . i) + (x . i) is V55() real ext-real Element of REAL
i is set
k is set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
(k,M,x) is Relation-like k -defined Function-like total set
i is set
D . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
x . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(M . i) -' (x . i) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
i is set
dom x is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
dom M is Element of bool k
dom D is Element of bool k
D . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
0 -' 0 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
0 -' (x . i) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(M . i) -' (x . i) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
i is set
k is set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
(k,D,M) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
(k,D,M) is Relation-like k -defined Function-like total set
rng (k,D,M) is set
i is set
dom (k,D,M) is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
Fg is set
(k,D,M) . Fg is set
D . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(D . Fg) -' (M . Fg) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is set
D is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
M is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
(k,D,M) is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
x is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
(k,(k,D,M),x) is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
(k,M,x) is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
(k,D,(k,M,x)) is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
i is set
(k,(k,D,M),x) . i is V55() real ext-real set
(k,D,M) . i is V55() real ext-real set
x . i is V55() real ext-real set
((k,D,M) . i) + (x . i) is V55() real ext-real Element of REAL
D . i is V55() real ext-real set
M . i is V55() real ext-real set
(D . i) + (M . i) is V55() real ext-real Element of REAL
((D . i) + (M . i)) + (x . i) is V55() real ext-real Element of REAL
(M . i) + (x . i) is V55() real ext-real Element of REAL
(D . i) + ((M . i) + (x . i)) is V55() real ext-real Element of REAL
(k,M,x) . i is V55() real ext-real set
(D . i) + ((k,M,x) . i) is V55() real ext-real Element of REAL
(k,D,(k,M,x)) . i is V55() real ext-real set
k is set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
(k,D,M) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
(k,(k,D,M),x) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
(k,M,x) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
(k,D,(k,M,x)) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
i is set
(k,(k,D,M),x) . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(k,D,M) . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
x . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
((k,D,M) . i) -' (x . i) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
D . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(D . i) -' (M . i) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((D . i) -' (M . i)) -' (x . i) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(M . i) + (x . i) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(D . i) -' ((M . i) + (x . i)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,M,x) . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(D . i) -' ((k,M,x) . i) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is Relation-like Function-like set
dom k is set
D is set
M is set
k . M is set
D is set
M is set
x is set
k . x is set
k is Relation-like Function-like set
(k) is set
dom k is set
D is set
k . D is set
k is Relation-like Function-like set
dom k is set
(k) is set
{0} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
0 .--> 1 is Relation-like NAT -defined {0} -defined NAT -valued RAT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued () set
{0} --> 1 is Relation-like non-empty {0} -defined NAT -valued RAT -valued {1} -valued Function-like constant non empty total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued () Element of bool [:{0},{1}:]
[:{0},{1}:] is Relation-like non empty finite countable set
bool [:{0},{1}:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
k is Relation-like NAT -defined {0} -defined NAT -valued RAT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued () set
k is Relation-like Function-like () set
(k) is set
k is set
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
{0} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
[:k,{0}:] is Relation-like set
dom (k --> 0) is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
M is Relation-like k -defined NAT -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
(M) is set
dom M is Element of bool k
x is set
M . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
k is Relation-like Function-like () set
D is set
M is set
k +* (D,M) is Relation-like Function-like set
((k +* (D,M))) is set
(k) is finite countable set
{D} is non empty trivial finite 1 -element countable set
(k) \/ {D} is non empty finite countable set
i is set
(k +* (D,M)) . i is set
dom k is set
dom k is set
k . i is set
dom k is set
k . i is set
dom k is set
k is set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
{0} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
[:k,{0}:] is Relation-like set
dom (k --> 0) is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
M is Relation-like k -defined Function-like total set
(M) is set
x is set
dom M is Element of bool k
M . x is set
k is set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
(k,D,M) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
((k,D,M)) is set
(D) is set
(M) is set
(D) \/ (M) is set
x is set
D . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(D . x) + (M . x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,D,M) . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
k is set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
(k,D,M) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
((k,D,M)) is set
(D) is set
x is set
D . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(D . x) -' (M . x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,D,M) . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
k is finite countable set
D is Relation-like k -defined Function-like total set
(D) is set
dom D is finite countable Element of bool k
bool k is non empty cup-closed diff-closed preBoolean finite V40() countable set
k is set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D,M) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
((k,D,M)) is set
(D) is finite countable set
(M) is finite countable set
(D) \/ (M) is finite countable set
(k,D,M) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
((k,D,M)) is set
(D) is finite countable set
k is set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
{0} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
[:k,{0}:] is Relation-like set
dom (k --> 0) is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
((k --> 0)) is set
M is set
(k --> 0) . M is Relation-like epsilon-transitive epsilon-connected ordinal natural Function-like finite cardinal countable V55() real ext-real non negative V61() () set
k is epsilon-transitive epsilon-connected ordinal set
i is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
Fg is epsilon-transitive epsilon-connected ordinal set
i . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
x . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
db is epsilon-transitive epsilon-connected ordinal set
x . db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
i . db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
k is epsilon-transitive epsilon-connected ordinal set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
i is epsilon-transitive epsilon-connected ordinal set
M . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
Fg is epsilon-transitive epsilon-connected ordinal set
x . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
i /\ Fg is epsilon-transitive epsilon-connected ordinal set
db is epsilon-transitive epsilon-connected ordinal set
x . db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
k1 is epsilon-transitive epsilon-connected ordinal set
D . k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
x . k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . k1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
k is epsilon-transitive epsilon-connected ordinal set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k is epsilon-transitive epsilon-connected ordinal set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k is epsilon-transitive epsilon-connected ordinal set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k is epsilon-transitive epsilon-connected ordinal set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k is epsilon-transitive epsilon-connected ordinal set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is set
D . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
i is epsilon-transitive epsilon-connected ordinal set
D . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
Fg is epsilon-transitive epsilon-connected ordinal set
M . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
k is set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
i is set
x . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
k is set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is set
D . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
dom D is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
dom D is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
dom D is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
k is epsilon-transitive epsilon-connected ordinal set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M,D) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,(k,M,D),D) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is set
i is epsilon-transitive epsilon-connected ordinal set
D . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(k,(k,M,D),D) . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(k,M,D) . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
((k,M,D) . x) + (D . x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(M . x) -' (D . x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((M . x) -' (D . x)) + (D . x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(M . x) + (D . x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((M . x) + (D . x)) -' (D . x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M,D) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,(k,M,D),D) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is set
(k,(k,M,D),D) . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(k,M,D) . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
((k,M,D) . x) -' (D . x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(M . x) + (D . x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((M . x) + (D . x)) -' (D . x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is epsilon-transitive epsilon-connected ordinal set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is set
D . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
i is epsilon-transitive epsilon-connected ordinal set
D . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
Fg is epsilon-transitive epsilon-connected ordinal set
M . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
db is epsilon-transitive epsilon-connected ordinal set
M . db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
k is set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M,x) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
i is set
D . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
x . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(M . i) + (x . i) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is set
Funcs (k,NAT) is functional non empty FUNCTION_DOMAIN of k, NAT
D is set
M is set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
dom x is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
rng x is V72() V73() V74() V75() V77() Element of bool RAT
bool RAT is non empty cup-closed diff-closed preBoolean set
D is set
M is set
x is set
k is set
(k) is set
bool (k) is non empty cup-closed diff-closed preBoolean set
({}) is Element of bool ({})
({}) is set
bool ({}) is non empty cup-closed diff-closed preBoolean set
k is set
D is Relation-like non-empty empty-yielding {} -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty total finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () () set
k is set
(k) is Element of bool (k)
(k) is set
bool (k) is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
{0} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
[:k,{0}:] is Relation-like set
k is set
(k) is non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
bool (k) is non empty cup-closed diff-closed preBoolean set
D is Element of bool (k)
M is set
k is set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
bool (k) is non empty cup-closed diff-closed preBoolean set
D is functional Element of bool (k)
M is Relation-like Function-like Element of D
x is functional non empty Element of bool (k)
i is Relation-like Function-like Element of x
k is set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
bool (k) is non empty cup-closed diff-closed preBoolean set
D is functional non empty Element of bool (k)
M is Relation-like k -defined Function-like Element of D
x is Relation-like k -defined Function-like Element of D
i is Relation-like k -defined Function-like Element of D
k is set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
{0} is functional non empty trivial finite V40() 1 -element with_common_domain countable set
[:k,{0}:] is Relation-like set
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
k is set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is set
(k) . D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
dom (k --> 0) is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
k is set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D,(k)) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is set
(k) . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(k,D,(k)) . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(D . M) + ((k) . M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D,(k)) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is set
(k) . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(k,D,(k)) . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(D . M) -' ((k) . M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,(k),D) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is set
(k) . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(k,(k),D) . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
((k) . M) -' (D . M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D,D) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is set
(k,D,D) . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(D . M) -' (D . M) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k) . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
k is set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M,D) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is set
(k,M,D) . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(M . x) -' (D . x) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is set
(k) . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
k is set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is set
(k) . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
k is epsilon-transitive epsilon-connected ordinal set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued T-Sequence-like Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k is epsilon-transitive epsilon-connected ordinal set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
[:(k),(k):] is Relation-like non empty set
bool [:(k),(k):] is non empty cup-closed diff-closed preBoolean set
D is Relation-like (k) -defined (k) -valued Element of bool [:(k),(k):]
M is set
x is set
[M,x] is set
{M,x} is non empty finite countable set
{M} is non empty trivial finite 1 -element countable set
{{M,x},{M}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
[x,M] is set
{x,M} is non empty finite countable set
{x} is non empty trivial finite 1 -element countable set
{{x,M},{x}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
i is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
db is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
k1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
M is set
[M,M] is set
{M,M} is non empty finite countable set
{M} is non empty trivial finite 1 -element countable set
{{M,M},{M}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
dom D is functional Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
field D is set
dom D is set
rng D is set
(dom D) \/ (rng D) is set
M is set
x is set
i is set
[M,x] is set
{M,x} is non empty finite countable set
{M} is non empty trivial finite 1 -element countable set
{{M,x},{M}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
[x,i] is set
{x,i} is non empty finite countable set
{x} is non empty trivial finite 1 -element countable set
{{x,i},{x}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
[M,i] is set
{M,i} is non empty finite countable set
{{M,i},{M}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
db is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
k1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
k2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
i1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
j1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive Element of bool [:(k),(k):]
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
i is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
[x,i] is set
{x,i} is functional non empty finite countable set
{x} is functional non empty trivial finite 1 -element with_common_domain countable set
{{x,i},{x}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
db is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
D is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive Element of bool [:(k),(k):]
M is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive Element of bool [:(k),(k):]
x is set
i is set
[x,i] is set
{x,i} is non empty finite countable set
{x} is non empty trivial finite 1 -element countable set
{{x,i},{x}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
Fg is set
db is set
[Fg,db] is set
{Fg,db} is non empty finite countable set
{Fg} is non empty trivial finite 1 -element countable set
{{Fg,db},{Fg}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
k1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
Fg is set
db is set
[Fg,db] is set
{Fg,db} is non empty finite countable set
{Fg} is non empty trivial finite 1 -element countable set
{{Fg,db},{Fg}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
k1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k is epsilon-transitive epsilon-connected ordinal set
(k) is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive Element of bool [:(k),(k):]
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
[:(k),(k):] is Relation-like non empty set
bool [:(k),(k):] is non empty cup-closed diff-closed preBoolean set
D is set
[D,D] is set
{D,D} is non empty finite countable set
{D} is non empty trivial finite 1 -element countable set
{{D,D},{D}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k is epsilon-transitive epsilon-connected ordinal set
(k) is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive Element of bool [:(k),(k):]
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
[:(k),(k):] is Relation-like non empty set
bool [:(k),(k):] is non empty cup-closed diff-closed preBoolean set
M is set
x is set
[M,x] is set
{M,x} is non empty finite countable set
{M} is non empty trivial finite 1 -element countable set
{{M,x},{M}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
[x,M] is set
{x,M} is non empty finite countable set
{x} is non empty trivial finite 1 -element countable set
{{x,M},{x}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
i is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
db is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k is epsilon-transitive epsilon-connected ordinal set
(k) is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive Element of bool [:(k),(k):]
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
[:(k),(k):] is Relation-like non empty set
bool [:(k),(k):] is non empty cup-closed diff-closed preBoolean set
M is set
x is set
i is set
[M,x] is set
{M,x} is non empty finite countable set
{M} is non empty trivial finite 1 -element countable set
{{M,x},{M}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
[x,i] is set
{x,i} is non empty finite countable set
{x} is non empty trivial finite 1 -element countable set
{{x,i},{x}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
[M,i] is set
{M,i} is non empty finite countable set
{{M,i},{M}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
k1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
db is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
i1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
j1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k is epsilon-transitive epsilon-connected ordinal set
(k) is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive Element of bool [:(k),(k):]
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
[:(k),(k):] is Relation-like non empty set
bool [:(k),(k):] is non empty cup-closed diff-closed preBoolean set
M is set
x is set
[M,x] is set
{M,x} is non empty finite countable set
{M} is non empty trivial finite 1 -element countable set
{{M,x},{M}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
[x,M] is set
{x,M} is non empty finite countable set
{x} is non empty trivial finite 1 -element countable set
{{x,M},{x}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
i is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k is epsilon-transitive epsilon-connected ordinal set
(k) is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive Element of bool [:(k),(k):]
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
[:(k),(k):] is Relation-like non empty set
bool [:(k),(k):] is non empty cup-closed diff-closed preBoolean set
field (k) is set
dom (k) is set
rng (k) is set
(dom (k)) \/ (rng (k)) is set
k is set
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
Funcs (k,NAT) is functional non empty FUNCTION_DOMAIN of k, NAT
bool (Funcs (k,NAT)) is non empty cup-closed diff-closed preBoolean set
D is Relation-like k -defined NAT -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
M is functional Element of bool (Funcs (k,NAT))
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
i is set
x . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
dom x is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
rng x is V72() V73() V74() V75() V77() Element of bool RAT
bool RAT is non empty cup-closed diff-closed preBoolean set
M is functional Element of bool (Funcs (k,NAT))
x is functional Element of bool (Funcs (k,NAT))
i is Relation-like k -defined NAT -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of Funcs (k,NAT)
Fg is set
i . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
D . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
Fg is set
i . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
D . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
k is set
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
D is Relation-like k -defined NAT -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
(k,D) is functional Element of bool (Funcs (k,NAT))
Funcs (k,NAT) is functional non empty FUNCTION_DOMAIN of k, NAT
bool (Funcs (k,NAT)) is non empty cup-closed diff-closed preBoolean set
M is set
D . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
k is set
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
D is Relation-like k -defined NAT -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
(k,D) is functional Element of bool (Funcs (k,NAT))
Funcs (k,NAT) is functional non empty FUNCTION_DOMAIN of k, NAT
bool (Funcs (k,NAT)) is non empty cup-closed diff-closed preBoolean set
k is set
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
Funcs (k,NAT) is functional non empty FUNCTION_DOMAIN of k, NAT
D is Relation-like k -defined NAT -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
(k,D) is functional non empty Element of bool (Funcs (k,NAT))
bool (Funcs (k,NAT)) is non empty cup-closed diff-closed preBoolean set
M is Relation-like Function-like Element of (k,D)
rng M is set
k is set
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
D is Relation-like k -defined NAT -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued () Element of bool [:k,NAT:]
(k,D) is functional non empty Element of bool (Funcs (k,NAT))
Funcs (k,NAT) is functional non empty FUNCTION_DOMAIN of k, NAT
bool (Funcs (k,NAT)) is non empty cup-closed diff-closed preBoolean set
M is set
x is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of (k,D)
dom x is set
(x) is set
(D) is finite countable set
i is set
x . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
k is set
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
D is Relation-like k -defined NAT -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued () Element of bool [:k,NAT:]
(D) is finite countable set
Fin k is non empty cup-closed diff-closed preBoolean set
dom D is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
k is non empty set
[:k,NAT:] is Relation-like non empty non trivial non finite non empty-membered set
bool [:k,NAT:] is non empty non trivial cup-closed diff-closed preBoolean non finite non empty-membered set
Fin k is non empty cup-closed diff-closed preBoolean set
D is Relation-like k -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued () Element of bool [:k,NAT:]
(k,D) is functional non empty Element of bool (Funcs (k,NAT))
Funcs (k,NAT) is functional non empty FUNCTION_DOMAIN of k, NAT
bool (Funcs (k,NAT)) is non empty cup-closed diff-closed preBoolean set
card (k,D) is epsilon-transitive epsilon-connected ordinal non empty cardinal set
(k,D) is finite countable Element of Fin k
addnat [:] (D,1) is Relation-like k -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
dom D is non empty set
(dom D) --> 1 is Relation-like non-empty dom D -defined NAT -valued RAT -valued Function-like constant non empty total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued set
[:(dom D),{1}:] is Relation-like non empty set
<:D,((dom D) --> 1):> is Relation-like Function-like set
<:D,((dom D) --> 1):> * addnat is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
multnat $$ ((k,D),(addnat [:] (D,1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M is Element of k
D . M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M is finite countable Element of Fin k
x is Element of k
{.x.} is non empty trivial finite 1 -element countable Element of Fin k
M \/ {.x.} is non empty finite countable Element of Fin k
i is Relation-like k -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
(k,i) is functional non empty Element of bool (Funcs (k,NAT))
card (k,i) is epsilon-transitive epsilon-connected ordinal non empty cardinal set
addnat [:] (i,1) is Relation-like k -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
dom i is non empty set
(dom i) --> 1 is Relation-like non-empty dom i -defined NAT -valued RAT -valued Function-like constant non empty total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued set
[:(dom i),{1}:] is Relation-like non empty set
<:i,((dom i) --> 1):> is Relation-like Function-like set
<:i,((dom i) --> 1):> * addnat is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
multnat $$ ((M \/ {.x.}),(addnat [:] (i,1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
{x} is non empty trivial finite 1 -element countable Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
M \/ {x} is non empty finite countable set
i +* (x,0) is Relation-like k -defined NAT -valued RAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
(i +* (x,0)) | M is Relation-like k -defined M -defined k -defined NAT -valued RAT -valued Function-like finite countable complex-valued ext-real-valued real-valued natural-valued () Element of bool [:k,NAT:]
i | M is Relation-like k -defined M -defined k -defined NAT -valued RAT -valued Function-like finite countable complex-valued ext-real-valued real-valued natural-valued () Element of bool [:k,NAT:]
addnat [:] ((i +* (x,0)),1) is Relation-like k -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
dom (i +* (x,0)) is non empty set
(dom (i +* (x,0))) --> 1 is Relation-like non-empty dom (i +* (x,0)) -defined NAT -valued RAT -valued Function-like constant non empty total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued set
[:(dom (i +* (x,0))),{1}:] is Relation-like non empty set
<:(i +* (x,0)),((dom (i +* (x,0))) --> 1):> is Relation-like Function-like set
<:(i +* (x,0)),((dom (i +* (x,0))) --> 1):> * addnat is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(addnat [:] ((i +* (x,0)),1)) | M is Relation-like k -defined M -defined k -defined NAT -valued RAT -valued Function-like finite countable complex-valued ext-real-valued real-valued natural-valued () Element of bool [:k,NAT:]
(addnat [:] (i,1)) | M is Relation-like k -defined M -defined k -defined NAT -valued RAT -valued Function-like finite countable complex-valued ext-real-valued real-valued natural-valued () Element of bool [:k,NAT:]
multnat $$ (M,(addnat [:] ((i +* (x,0)),1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
multnat $$ (M,(addnat [:] (i,1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom i is non empty Element of bool k
db is Element of k
(i +* (x,0)) . db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() Element of NAT
i . db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,(i +* (x,0))) is functional non empty Element of bool (Funcs (k,NAT))
card (k,(i +* (x,0))) is epsilon-transitive epsilon-connected ordinal non empty cardinal set
i . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(i . x) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
dom (addnat [:] (i,1)) is non empty Element of bool k
(addnat [:] (i,1)) . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
addnat . ((i . x),1) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
[(i . x),1] is set
{(i . x),1} is non empty finite V40() countable set
{(i . x)} is non empty trivial finite V40() 1 -element countable set
{{(i . x),1},{(i . x)}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
addnat . [(i . x),1] is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
db is functional non empty finite countable set
card db is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
[:db,((i . x) + 1):] is Relation-like non empty finite countable set
k1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
i1 is Relation-like Function-like Element of db
j1 is epsilon-transitive epsilon-connected ordinal Element of k1
i1 +* (x,j1) is Relation-like Function-like set
dom i1 is set
dom (i1 +* (x,j1)) is set
dbi1 is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of (k,(i +* (x,0)))
ddbi11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dbi1 +* (x,ddbi11) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
b11 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
b12 is set
b11 . b12 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
i . b12 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
dbi1 . b12 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(i +* (x,0)) . b12 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
[:db,k1:] is Relation-like non empty finite countable set
[:[:db,k1:],(k,i):] is Relation-like non empty set
bool [:[:db,k1:],(k,i):] is non empty cup-closed diff-closed preBoolean set
i1 is Relation-like [:db,k1:] -defined (k,i) -valued Function-like non empty total quasi_total finite countable Function-yielding V50() () Element of bool [:[:db,k1:],(k,i):]
dom i1 is Relation-like non empty finite countable set
rng i1 is non empty finite countable set
j1 is set
dbi1 is set
i1 . j1 is Relation-like Function-like set
i1 . dbi1 is Relation-like Function-like set
dom i1 is Relation-like db -defined k1 -valued non empty finite countable Element of bool [:db,k1:]
bool [:db,k1:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
ddbi11 is set
b11 is set
[ddbi11,b11] is set
{ddbi11,b11} is non empty finite countable set
{ddbi11} is non empty trivial finite 1 -element countable set
{{ddbi11,b11},{ddbi11}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
b12 is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of (k,(i +* (x,0)))
dom b12 is set
b119 is set
b129 is set
[b119,b129] is set
{b119,b129} is non empty finite countable set
{b119} is non empty trivial finite 1 -element countable set
{{b119,b129},{b119}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
b111 is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of (k,(i +* (x,0)))
dom b111 is set
b111 +* (x,b129) is Relation-like Function-like set
i1 . (b111,b129) is set
[b111,b129] is set
{b111,b129} is non empty finite countable set
{b111} is functional non empty trivial finite 1 -element with_common_domain countable set
{{b111,b129},{b111}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
i1 . [b111,b129] is Relation-like Function-like set
i1 . (b12,b11) is set
[b12,b11] is set
{b12,b11} is non empty finite countable set
{b12} is functional non empty trivial finite 1 -element with_common_domain countable set
{{b12,b11},{b12}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
i1 . [b12,b11] is Relation-like Function-like set
b12 +* (x,b11) is Relation-like Function-like set
j2 is set
(i +* (x,0)) . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() Element of NAT
b112 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
b112 . j2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
i2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
i2 . j2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
b112 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
b112 . j2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(b111 +* (x,b129)) . j2 is set
i2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
i2 . j2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
b112 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
i2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
b112 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
i2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
(b111 +* (x,b129)) . x is set
dom i1 is Relation-like db -defined k1 -valued non empty finite countable Element of bool [:db,k1:]
bool [:db,k1:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
rng i1 is functional non empty finite countable Element of bool (k,i)
bool (k,i) is non empty cup-closed diff-closed preBoolean set
j1 is set
dbi1 is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of (k,i)
dom dbi1 is set
dbi1 +* (x,0) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom (dbi1 +* (x,0)) is set
b11 is set
ddbi11 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
ddbi11 . b11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
(i +* (x,0)) . b11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
ddbi11 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
ddbi11 . b11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
dbi1 . b11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
i . b11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
(i +* (x,0)) . b11 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
ddbi11 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
ddbi11 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued set
dbi1 . x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
b11 is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of (k,(i +* (x,0)))
[b11,(dbi1 . x)] is set
{b11,(dbi1 . x)} is non empty finite countable set
{b11} is functional non empty trivial finite 1 -element with_common_domain countable set
{{b11,(dbi1 . x)},{b11}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
dbi1 +* (x,(dbi1 . x)) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
b11 +* (x,(dbi1 . x)) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
i1 . (b11,(dbi1 . x)) is set
i1 . [b11,(dbi1 . x)] is Relation-like Function-like set
card [:db,((i . x) + 1):] is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
card ((i . x) + 1) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(card db) * (card ((i . x) + 1)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(card db) * ((i . x) + 1) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
multnat . ((multnat $$ (M,(addnat [:] (i,1)))),((i . x) + 1)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
[(multnat $$ (M,(addnat [:] (i,1)))),((i . x) + 1)] is set
{(multnat $$ (M,(addnat [:] (i,1)))),((i . x) + 1)} is non empty finite V40() countable set
{(multnat $$ (M,(addnat [:] (i,1))))} is non empty trivial finite V40() 1 -element countable set
{{(multnat $$ (M,(addnat [:] (i,1)))),((i . x) + 1)},{(multnat $$ (M,(addnat [:] (i,1))))}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
multnat . [(multnat $$ (M,(addnat [:] (i,1)))),((i . x) + 1)] is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
{}. k is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty finite finite-yielding V40() cardinal {} -element Cardinal-yielding countable Function-yielding V50() FinSequence-like FinSubsequence-like FinSequence-membered V55() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued () () Element of Fin k
M is Relation-like k -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
(k,M) is functional non empty Element of bool (Funcs (k,NAT))
card (k,M) is epsilon-transitive epsilon-connected ordinal non empty cardinal set
addnat [:] (M,1) is Relation-like k -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
dom M is non empty set
(dom M) --> 1 is Relation-like non-empty dom M -defined NAT -valued RAT -valued Function-like constant non empty total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued set
[:(dom M),{1}:] is Relation-like non empty set
<:M,((dom M) --> 1):> is Relation-like Function-like set
<:M,((dom M) --> 1):> * addnat is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
multnat $$ (({}. k),(addnat [:] (M,1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x is set
Fg is Element of k
M . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
i is Relation-like k -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
i . Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
{M} is functional non empty trivial finite 1 -element with_common_domain countable with_non-empty_elements non empty-membered Element of bool (bool [:k,NAT:])
bool (bool [:k,NAT:]) is non empty non trivial cup-closed diff-closed preBoolean non finite non empty-membered set
k is set
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
D is Relation-like k -defined NAT -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued () Element of bool [:k,NAT:]
(k,D) is functional non empty Element of bool (Funcs (k,NAT))
Funcs (k,NAT) is functional non empty FUNCTION_DOMAIN of k, NAT
bool (Funcs (k,NAT)) is non empty cup-closed diff-closed preBoolean set
Funcs ({},NAT) is functional non empty FUNCTION_DOMAIN of {} , NAT
M is non empty set
[:M,NAT:] is Relation-like non empty non trivial non finite non empty-membered set
bool [:M,NAT:] is non empty non trivial cup-closed diff-closed preBoolean non finite non empty-membered set
x is Relation-like M -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued () Element of bool [:M,NAT:]
(M,x) is functional non empty Element of bool (Funcs (M,NAT))
Funcs (M,NAT) is functional non empty FUNCTION_DOMAIN of M, NAT
bool (Funcs (M,NAT)) is non empty cup-closed diff-closed preBoolean set
card (M,x) is epsilon-transitive epsilon-connected ordinal non empty cardinal set
(M,x) is finite countable Element of Fin M
Fin M is non empty cup-closed diff-closed preBoolean set
addnat [:] (x,1) is Relation-like M -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:M,NAT:]
dom x is non empty set
(dom x) --> 1 is Relation-like non-empty dom x -defined NAT -valued RAT -valued Function-like constant non empty total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued set
[:(dom x),{1}:] is Relation-like non empty set
<:x,((dom x) --> 1):> is Relation-like Function-like set
<:x,((dom x) --> 1):> * addnat is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
multnat $$ ((M,x),(addnat [:] (x,1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is epsilon-transitive epsilon-connected ordinal set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
bool (k) is non empty cup-closed diff-closed preBoolean set
(k) is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive being_linear-order Element of bool [:(k),(k):]
[:(k),(k):] is Relation-like non empty set
bool [:(k),(k):] is non empty cup-closed diff-closed preBoolean set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
dom D is Element of bool k
bool k is non empty cup-closed diff-closed preBoolean set
rng D is V72() V73() V74() V75() V77() Element of bool RAT
bool RAT is non empty cup-closed diff-closed preBoolean set
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
M is Relation-like k -defined NAT -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued () Element of bool [:k,NAT:]
(k,M) is functional non empty finite countable Element of bool (Funcs (k,NAT))
Funcs (k,NAT) is functional non empty FUNCTION_DOMAIN of k, NAT
bool (Funcs (k,NAT)) is non empty cup-closed diff-closed preBoolean set
x is functional non empty finite countable Element of bool (k)
((k),x,(k)) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
i is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
db is set
Fg . db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
db is set
Fg . db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
D . db is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative V61() set
M is Relation-like NAT -defined (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
x is Relation-like NAT -defined (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
i is functional non empty finite countable Element of bool (k)
((k),i,(k)) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
Fg is functional non empty finite countable Element of bool (k)
((k),Fg,(k)) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
db is set
k1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
k1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
k is epsilon-transitive epsilon-connected ordinal set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D) is Relation-like NAT -defined (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
bool (k) is non empty cup-closed diff-closed preBoolean set
(k) is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive being_linear-order Element of bool [:(k),(k):]
[:(k),(k):] is Relation-like non empty set
bool [:(k),(k):] is non empty cup-closed diff-closed preBoolean set
M is functional non empty finite countable Element of bool (k)
((k),M,(k)) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
k is epsilon-transitive epsilon-connected ordinal set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
dom (k,M) is non empty finite countable V77() Element of bool NAT
(k,M) /. D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k,M) . D is Relation-like Function-like set
rng (k,M) is functional non empty finite countable Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
(k) is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive being_linear-order Element of bool [:(k),(k):]
[:(k),(k):] is Relation-like non empty set
bool [:(k),(k):] is non empty cup-closed diff-closed preBoolean set
i is functional non empty finite countable Element of bool (k)
((k),i,(k)) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
k is epsilon-transitive epsilon-connected ordinal set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
k --> 0 is Relation-like k -defined NAT -valued RAT -valued T-Sequence-like Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
(k,D) /. 1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
len (k,D) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(k,D) /. (len (k,D)) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
(k) is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive being_linear-order Element of bool [:(k),(k):]
[:(k),(k):] is Relation-like non empty set
bool [:(k),(k):] is non empty cup-closed diff-closed preBoolean set
M is functional non empty finite countable Element of bool (k)
((k),M,(k)) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
[x,D] is set
{x,D} is functional non empty finite countable set
{x} is functional non empty trivial finite 1 -element with_common_domain countable set
{{x,D},{x}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
[(k),x] is Element of [:(k),(k):]
{(k),x} is functional non empty finite countable set
{(k)} is functional non empty trivial finite 1 -element with_common_domain countable set
{{(k),x},{(k)}} is non empty finite V40() countable with_non-empty_elements non empty-membered set
k is epsilon-transitive epsilon-connected ordinal set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
k --> 0 is Relation-like k -defined NAT -valued RAT -valued T-Sequence-like Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
len (k,M) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
(k,M) /. D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
dom (k,M) is non empty finite countable V77() Element of bool NAT
(k,M) /. 1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k,M) /. (len (k,M)) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
k is epsilon-transitive epsilon-connected ordinal set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued T-Sequence-like Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
(k,(k)) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
((k),(k)) is Relation-like NAT -defined (k) -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
(k) * is functional non empty FinSequence-membered FinSequenceSet of (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
(k) is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive being_linear-order Element of bool [:(k),(k):]
[:(k),(k):] is Relation-like non empty set
bool [:(k),(k):] is non empty cup-closed diff-closed preBoolean set
D is functional non empty finite countable Element of bool (k)
((k),D,(k)) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
{(k)} is functional non empty trivial finite 1 -element with_common_domain countable Element of bool (k)
M is set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
rng (k,(k)) is functional non empty finite countable Element of bool (k)
len (k,(k)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
card (rng (k,(k))) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
k is epsilon-transitive epsilon-connected ordinal set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
2 -tuples_on (k) is functional non empty FinSequence-membered FinSequenceSet of (k)
(k) * is functional non empty FinSequence-membered FinSequenceSet of (k)
{ b1 where b1 is Relation-like NAT -defined (k) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of (k) * : len b1 = 2 } is set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
dom (k,D) is non empty finite countable V77() Element of bool NAT
len (k,D) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
Seg (len (k,D)) is non empty finite len (k,D) -element countable Element of bool NAT
M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
(k,D) /. M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D,x) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*x,(k,D,x)*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
len <*x,(k,D,x)*> is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
i is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),i,Fg) is Relation-like NAT -defined (k) -valued Function-like non empty finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
db is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
k1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D,k1) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*k1,(k,D,k1)*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
M is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of 2 -tuples_on (k)
len M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom M is finite countable V77() Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,D) /. x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),NAT,(2 -tuples_on (k)),M,x) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
i is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D,i) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*i,(k,D,i)*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
M is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of 2 -tuples_on (k)
dom M is finite countable V77() Element of bool NAT
x is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of 2 -tuples_on (k)
dom x is finite countable V77() Element of bool NAT
i is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
(k,D) /. i is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),NAT,(2 -tuples_on (k)),M,i) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D,Fg) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*Fg,(k,D,Fg)*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
((k),NAT,(2 -tuples_on (k)),x,i) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
k is epsilon-transitive epsilon-connected ordinal set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
2 -tuples_on (k) is functional non empty FinSequence-membered FinSequenceSet of (k)
(k) * is functional non empty FinSequence-membered FinSequenceSet of (k)
{ b1 where b1 is Relation-like NAT -defined (k) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of (k) * : len b1 = 2 } is set
D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of 2 -tuples_on (k)
dom (k,M) is finite countable V77() Element of bool NAT
((k),NAT,(2 -tuples_on (k)),(k,M),D) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
(k,M) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
(k,M) /. D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M,x) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*x,(k,M,x)*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,x,(k,M,x)) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
dom (k,M) is non empty finite countable V77() Element of bool NAT
k is epsilon-transitive epsilon-connected ordinal set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
2 -tuples_on (k) is functional non empty FinSequence-membered FinSequenceSet of (k)
(k) * is functional non empty FinSequence-membered FinSequenceSet of (k)
{ b1 where b1 is Relation-like NAT -defined (k) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of (k) * : len b1 = 2 } is set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M,x) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of 2 -tuples_on (k)
dom (k,D) is finite countable V77() Element of bool NAT
<*M,x*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
bool (k) is non empty cup-closed diff-closed preBoolean set
(k,D) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
(k) is Relation-like (k) -defined (k) -valued total quasi_total reflexive antisymmetric transitive being_linear-order Element of bool [:(k),(k):]
[:(k),(k):] is Relation-like non empty set
bool [:(k),(k):] is non empty cup-closed diff-closed preBoolean set
i is functional non empty finite countable Element of bool (k)
((k),i,(k)) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
rng (k,D) is functional non empty finite countable Element of bool (k)
dom (k,D) is non empty finite countable V77() Element of bool NAT
Fg is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,D) /. Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),NAT,(2 -tuples_on (k)),(k,D),Fg) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
(k,D,M) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*M,(k,D,M)*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
k is epsilon-transitive epsilon-connected ordinal set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
2 -tuples_on (k) is functional non empty FinSequence-membered FinSequenceSet of (k)
(k) * is functional non empty FinSequence-membered FinSequenceSet of (k)
{ b1 where b1 is Relation-like NAT -defined (k) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of (k) * : len b1 = 2 } is set
D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of 2 -tuples_on (k)
dom (k,M) is finite countable V77() Element of bool NAT
((k),NAT,(2 -tuples_on (k)),(k,M),D) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
i is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*x,i*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,M) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
(k,M) /. D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M,Fg) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*Fg,(k,M,Fg)*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
k is epsilon-transitive epsilon-connected ordinal set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of 2 -tuples_on (k)
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
2 -tuples_on (k) is functional non empty FinSequence-membered FinSequenceSet of (k)
(k) * is functional non empty FinSequence-membered FinSequenceSet of (k)
{ b1 where b1 is Relation-like NAT -defined (k) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of (k) * : len b1 = 2 } is set
(k,D) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
dom (k,D) is non empty finite countable V77() Element of bool NAT
dom (k,D) is finite countable V77() Element of bool NAT
M is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((k),NAT,(2 -tuples_on (k)),(k,D),x) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
i is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*i,Fg*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,i,Fg) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
((k),NAT,(2 -tuples_on (k)),(k,D),M) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
(k,D) /. M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k,D) /. x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
M is set
dom (k,D) is finite countable V77() set
(k,D) . M is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,D) /. x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
k is epsilon-transitive epsilon-connected ordinal set
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k,D) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
2 -tuples_on (k) is functional non empty FinSequence-membered FinSequenceSet of (k)
(k) * is functional non empty FinSequence-membered FinSequenceSet of (k)
{ b1 where b1 is Relation-like NAT -defined (k) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of (k) * : len b1 = 2 } is set
k is epsilon-transitive epsilon-connected ordinal set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
2 -tuples_on (k) is functional non empty FinSequence-membered FinSequenceSet of (k)
(k) * is functional non empty FinSequence-membered FinSequenceSet of (k)
{ b1 where b1 is Relation-like NAT -defined (k) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of (k) * : len b1 = 2 } is set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
k --> 0 is Relation-like k -defined NAT -valued RAT -valued T-Sequence-like Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
((k),NAT,(2 -tuples_on (k)),(k,D),1) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
<*(k),D*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
len (k,D) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
((k),NAT,(2 -tuples_on (k)),(k,D),(len (k,D))) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
<*D,(k)*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,D) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
(k,D) /. 1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
dom (k,D) is non empty finite countable V77() Element of bool NAT
(k,D,(k)) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*(k),(k,D,(k))*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,D) /. (len (k,D)) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
dom (k,D) is non empty finite countable V77() Element of bool NAT
len (k,D) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D,D) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*D,(k,D,D)*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
k is epsilon-transitive epsilon-connected ordinal set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
2 -tuples_on (k) is functional non empty FinSequence-membered FinSequenceSet of (k)
(k) * is functional non empty FinSequence-membered FinSequenceSet of (k)
{ b1 where b1 is Relation-like NAT -defined (k) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of (k) * : len b1 = 2 } is set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
k --> 0 is Relation-like k -defined NAT -valued RAT -valued T-Sequence-like Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative set
M is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
len (k,M) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
((k),NAT,(2 -tuples_on (k)),(k,M),D) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
i is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*x,i*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,M) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
(k,M) /. D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
dom (k,M) is non empty finite countable V77() Element of bool NAT
Fg is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,M,Fg) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*Fg,(k,M,Fg)*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
dom (k,M) is non empty finite countable V77() Element of bool NAT
len (k,M) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
k is epsilon-transitive epsilon-connected ordinal set
(k) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k) is non empty set
(k) is functional non empty Element of bool (k)
bool (k) is non empty cup-closed diff-closed preBoolean set
k --> 0 is Relation-like k -defined NAT -valued RAT -valued T-Sequence-like Function-like constant total quasi_total Cardinal-yielding Function-yielding V50() complex-valued ext-real-valued real-valued natural-valued Element of bool [:k,NAT:]
[:k,NAT:] is Relation-like set
bool [:k,NAT:] is non empty cup-closed diff-closed preBoolean set
[:k,{0}:] is Relation-like set
(k,(k)) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
2 -tuples_on (k) is functional non empty FinSequence-membered FinSequenceSet of (k)
(k) * is functional non empty FinSequence-membered FinSequenceSet of (k)
{ b1 where b1 is Relation-like NAT -defined (k) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of (k) * : len b1 = 2 } is set
((k),(k),(k)) is Relation-like NAT -defined (k) -valued Function-like non empty finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
(((k) *),((k),(k),(k))) is Relation-like NAT -defined (k) * -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () () Element of ((k) *) *
((k) *) * is functional non empty FinSequence-membered FinSequenceSet of (k) *
len ((k),(k),(k)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
D is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
((2 -tuples_on (k)),D) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () () Element of (2 -tuples_on (k)) *
(2 -tuples_on (k)) * is functional non empty FinSequence-membered FinSequenceSet of 2 -tuples_on (k)
M is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of 2 -tuples_on (k)
dom M is finite countable V77() Element of bool NAT
((k),(k)) is Relation-like NAT -defined (k) -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
(k,(k)) is Relation-like NAT -defined (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of (k)
x is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(k,(k)) /. x is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),NAT,(2 -tuples_on (k)),M,x) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
i is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,(k),i) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*i,(k,(k),i)*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
dom (k,(k)) is non empty finite countable V77() Element of bool NAT
k is epsilon-transitive epsilon-connected ordinal set
(k) is functional non empty Element of bool (k)
(k) is non empty set
bool (k) is non empty cup-closed diff-closed preBoolean set
3 -tuples_on (k) is functional non empty FinSequence-membered FinSequenceSet of (k)
(k) * is functional non empty FinSequence-membered FinSequenceSet of (k)
{ b1 where b1 is Relation-like NAT -defined (k) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of (k) * : len b1 = 3 } is set
(3 -tuples_on (k)) * is functional non empty FinSequence-membered FinSequenceSet of 3 -tuples_on (k)
2 -tuples_on (k) is functional non empty FinSequence-membered FinSequenceSet of (k)
{ b1 where b1 is Relation-like NAT -defined (k) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of (k) * : len b1 = 2 } is set
D is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,D) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
dom (k,D) is non empty finite countable V77() Element of bool NAT
M is Relation-like NAT -defined (3 -tuples_on (k)) * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of (3 -tuples_on (k)) *
dom M is finite countable V77() Element of bool NAT
x is Relation-like NAT -defined (3 -tuples_on (k)) * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of (3 -tuples_on (k)) *
dom x is finite countable V77() Element of bool NAT
((3 -tuples_on (k)),M) is Relation-like NAT -defined 3 -tuples_on (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (3 -tuples_on (k)) *
dom ((3 -tuples_on (k)),M) is finite countable V77() Element of bool NAT
[:(dom ((3 -tuples_on (k)),M)),(dom ((3 -tuples_on (k)),M)):] is Relation-like RAT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
bool [:(dom ((3 -tuples_on (k)),M)),(dom ((3 -tuples_on (k)),M)):] is non empty cup-closed diff-closed preBoolean finite V40() countable set
((3 -tuples_on (k)),x) is Relation-like NAT -defined 3 -tuples_on (k) -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (3 -tuples_on (k)) *
k1 is set
dom ((3 -tuples_on (k)),x) is finite countable V77() set
k2 is set
((3 -tuples_on (k)),x) . k1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
((3 -tuples_on (k)),x) . k2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
dom ((3 -tuples_on (k)),x) is finite countable V77() Element of bool NAT
i1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x . i1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
dom (x . i1) is finite countable V77() Element of bool NAT
i1 -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x | (i1 -' 1) is Relation-like NAT -defined (3 -tuples_on (k)) * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of (3 -tuples_on (k)) *
((3 -tuples_on (k)),(x | (i1 -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued () FinSequence of NAT
Sum ((3 -tuples_on (k)),(x | (i1 -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum ((3 -tuples_on (k)),(x | (i1 -' 1)))) + j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(x . i1) . j1 is set
((k),NAT,(2 -tuples_on (k)),(k,D),i1) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
dbi1 is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
dbi1 /. 2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k,(dbi1 /. 2)) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
len (k,(dbi1 /. 2)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
dbi1 /. 1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),(dbi1 /. 1)) is Relation-like NAT -defined (k) -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
(len (k,(dbi1 /. 2))) |-> ((k),(dbi1 /. 1)) is Relation-like NAT -defined (k) * -valued Function-like finite len (k,(dbi1 /. 2)) -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () () Element of (len (k,(dbi1 /. 2))) -tuples_on ((k) *)
(len (k,(dbi1 /. 2))) -tuples_on ((k) *) is functional non empty FinSequence-membered FinSequenceSet of (k) *
((k) *) * is functional non empty FinSequence-membered FinSequenceSet of (k) *
{ b1 where b1 is Relation-like NAT -defined (k) * -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of ((k) *) * : len b1 = len (k,(dbi1 /. 2)) } is set
Seg (len (k,(dbi1 /. 2))) is non empty finite len (k,(dbi1 /. 2)) -element countable Element of bool NAT
(Seg (len (k,(dbi1 /. 2)))) --> ((k),(dbi1 /. 1)) is Relation-like non-empty Seg (len (k,(dbi1 /. 2))) -defined Seg (len (k,(dbi1 /. 2))) -defined (k) * -valued {((k),(dbi1 /. 1))} -valued Function-like constant non empty total total quasi_total finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of bool [:(Seg (len (k,(dbi1 /. 2)))),{((k),(dbi1 /. 1))}:]
{((k),(dbi1 /. 1))} is functional non empty trivial finite V40() 1 -element with_common_domain countable with_non-empty_elements non empty-membered set
[:(Seg (len (k,(dbi1 /. 2)))),{((k),(dbi1 /. 1))}:] is Relation-like non empty finite countable set
bool [:(Seg (len (k,(dbi1 /. 2)))),{((k),(dbi1 /. 1))}:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
(((len (k,(dbi1 /. 2))) |-> ((k),(dbi1 /. 1))),(k,(dbi1 /. 2))) is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () set
dom (k,(dbi1 /. 2)) is non empty finite countable V77() Element of bool NAT
dom ((len (k,(dbi1 /. 2))) |-> ((k),(dbi1 /. 1))) is finite len (k,(dbi1 /. 2)) -element countable V77() Element of bool NAT
(dom (k,(dbi1 /. 2))) /\ (dom ((len (k,(dbi1 /. 2))) |-> ((k),(dbi1 /. 1)))) is finite countable Element of bool NAT
(dom (k,(dbi1 /. 2))) /\ (Seg (len (k,(dbi1 /. 2)))) is finite countable Element of bool NAT
(dom (k,(dbi1 /. 2))) /\ (dom (k,(dbi1 /. 2))) is finite countable Element of bool NAT
((k),NAT,(2 -tuples_on (k)),(k,(dbi1 /. 2)),j1) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
(k,(dbi1 /. 2)) . j1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
((len (k,(dbi1 /. 2))) |-> ((k),(dbi1 /. 1))) . j1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
b11 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b12 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b11,b12*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,b11,b12) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b119 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
b129 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
b111 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b112 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b111,b112*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,b111,b112) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b11*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like () set
<*b11*> ^ <*b111,b112*> is Relation-like NAT -defined Function-like non empty finite 1 + 2 -element countable FinSequence-like FinSubsequence-like () set
1 + 2 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
<*b11,b111,b112*> is Relation-like NAT -defined Function-like non empty finite 3 -element countable FinSequence-like FinSubsequence-like () set
i2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
j2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x . i2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
dom (x . i2) is finite countable V77() Element of bool NAT
i2 -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x | (i2 -' 1) is Relation-like NAT -defined (3 -tuples_on (k)) * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of (3 -tuples_on (k)) *
((3 -tuples_on (k)),(x | (i2 -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued () FinSequence of NAT
Sum ((3 -tuples_on (k)),(x | (i2 -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum ((3 -tuples_on (k)),(x | (i2 -' 1)))) + j2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(x . i2) . j2 is set
((k),NAT,(2 -tuples_on (k)),(k,D),i2) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
dbi2 is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
dbi2 /. 2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k,(dbi2 /. 2)) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
len (k,(dbi2 /. 2)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
dbi2 /. 1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),(dbi2 /. 1)) is Relation-like NAT -defined (k) -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
(len (k,(dbi2 /. 2))) |-> ((k),(dbi2 /. 1)) is Relation-like NAT -defined (k) * -valued Function-like finite len (k,(dbi2 /. 2)) -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () () Element of (len (k,(dbi2 /. 2))) -tuples_on ((k) *)
(len (k,(dbi2 /. 2))) -tuples_on ((k) *) is functional non empty FinSequence-membered FinSequenceSet of (k) *
{ b1 where b1 is Relation-like NAT -defined (k) * -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of ((k) *) * : len b1 = len (k,(dbi2 /. 2)) } is set
Seg (len (k,(dbi2 /. 2))) is non empty finite len (k,(dbi2 /. 2)) -element countable Element of bool NAT
(Seg (len (k,(dbi2 /. 2)))) --> ((k),(dbi2 /. 1)) is Relation-like non-empty Seg (len (k,(dbi2 /. 2))) -defined Seg (len (k,(dbi2 /. 2))) -defined (k) * -valued {((k),(dbi2 /. 1))} -valued Function-like constant non empty total total quasi_total finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of bool [:(Seg (len (k,(dbi2 /. 2)))),{((k),(dbi2 /. 1))}:]
{((k),(dbi2 /. 1))} is functional non empty trivial finite V40() 1 -element with_common_domain countable with_non-empty_elements non empty-membered set
[:(Seg (len (k,(dbi2 /. 2)))),{((k),(dbi2 /. 1))}:] is Relation-like non empty finite countable set
bool [:(Seg (len (k,(dbi2 /. 2)))),{((k),(dbi2 /. 1))}:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
(((len (k,(dbi2 /. 2))) |-> ((k),(dbi2 /. 1))),(k,(dbi2 /. 2))) is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () set
dom (k,(dbi2 /. 2)) is non empty finite countable V77() Element of bool NAT
dom ((len (k,(dbi2 /. 2))) |-> ((k),(dbi2 /. 1))) is finite len (k,(dbi2 /. 2)) -element countable V77() Element of bool NAT
(dom (k,(dbi2 /. 2))) /\ (dom ((len (k,(dbi2 /. 2))) |-> ((k),(dbi2 /. 1)))) is finite countable Element of bool NAT
(dom (k,(dbi2 /. 2))) /\ (Seg (len (k,(dbi2 /. 2)))) is finite countable Element of bool NAT
(dom (k,(dbi2 /. 2))) /\ (dom (k,(dbi2 /. 2))) is finite countable Element of bool NAT
((k),NAT,(2 -tuples_on (k)),(k,(dbi2 /. 2)),j2) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
(k,(dbi2 /. 2)) . j2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
((len (k,(dbi2 /. 2))) |-> ((k),(dbi2 /. 1))) . j2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
b21 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b22 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b21,b22*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,b21,b22) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b219 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
b229 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
b211 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b212 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b211,b212*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,b211,b212) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b21*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like () set
<*b21*> ^ <*b211,b212*> is Relation-like NAT -defined Function-like non empty finite 1 + 2 -element countable FinSequence-like FinSubsequence-like () set
<*b21,b211,b212*> is Relation-like NAT -defined Function-like non empty finite 3 -element countable FinSequence-like FinSubsequence-like () set
(k,D) . i2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
(k,D) . i1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
k1 is set
rng ((3 -tuples_on (k)),M) is functional finite countable FinSequence-membered Element of bool (3 -tuples_on (k))
bool (3 -tuples_on (k)) is non empty cup-closed diff-closed preBoolean set
k2 is set
((3 -tuples_on (k)),M) . k2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
i1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((3 -tuples_on (k)),M) . i1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dbi1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M . j1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
dom (M . j1) is finite countable V77() Element of bool NAT
j1 -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M | (j1 -' 1) is Relation-like NAT -defined (3 -tuples_on (k)) * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of (3 -tuples_on (k)) *
((3 -tuples_on (k)),(M | (j1 -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued () FinSequence of NAT
Sum ((3 -tuples_on (k)),(M | (j1 -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum ((3 -tuples_on (k)),(M | (j1 -' 1)))) + dbi1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(M . j1) . dbi1 is set
((k),NAT,(2 -tuples_on (k)),(k,D),j1) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
ddbi11 is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
ddbi11 /. 1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k,(ddbi11 /. 1)) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
len (k,(ddbi11 /. 1)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
ddbi11 /. 2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),(ddbi11 /. 2)) is Relation-like NAT -defined (k) -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
(len (k,(ddbi11 /. 1))) |-> ((k),(ddbi11 /. 2)) is Relation-like NAT -defined (k) * -valued Function-like finite len (k,(ddbi11 /. 1)) -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () () Element of (len (k,(ddbi11 /. 1))) -tuples_on ((k) *)
(len (k,(ddbi11 /. 1))) -tuples_on ((k) *) is functional non empty FinSequence-membered FinSequenceSet of (k) *
((k) *) * is functional non empty FinSequence-membered FinSequenceSet of (k) *
{ b1 where b1 is Relation-like NAT -defined (k) * -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of ((k) *) * : len b1 = len (k,(ddbi11 /. 1)) } is set
Seg (len (k,(ddbi11 /. 1))) is non empty finite len (k,(ddbi11 /. 1)) -element countable Element of bool NAT
(Seg (len (k,(ddbi11 /. 1)))) --> ((k),(ddbi11 /. 2)) is Relation-like non-empty Seg (len (k,(ddbi11 /. 1))) -defined Seg (len (k,(ddbi11 /. 1))) -defined (k) * -valued {((k),(ddbi11 /. 2))} -valued Function-like constant non empty total total quasi_total finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of bool [:(Seg (len (k,(ddbi11 /. 1)))),{((k),(ddbi11 /. 2))}:]
{((k),(ddbi11 /. 2))} is functional non empty trivial finite V40() 1 -element with_common_domain countable with_non-empty_elements non empty-membered set
[:(Seg (len (k,(ddbi11 /. 1)))),{((k),(ddbi11 /. 2))}:] is Relation-like non empty finite countable set
bool [:(Seg (len (k,(ddbi11 /. 1)))),{((k),(ddbi11 /. 2))}:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
((k,(ddbi11 /. 1)),((len (k,(ddbi11 /. 1))) |-> ((k),(ddbi11 /. 2)))) is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () set
dom (k,(ddbi11 /. 1)) is non empty finite countable V77() Element of bool NAT
dom ((len (k,(ddbi11 /. 1))) |-> ((k),(ddbi11 /. 2))) is finite len (k,(ddbi11 /. 1)) -element countable V77() Element of bool NAT
(dom (k,(ddbi11 /. 1))) /\ (dom ((len (k,(ddbi11 /. 1))) |-> ((k),(ddbi11 /. 2)))) is finite countable Element of bool NAT
(dom (k,(ddbi11 /. 1))) /\ (Seg (len (k,(ddbi11 /. 1)))) is finite countable Element of bool NAT
(dom (k,(ddbi11 /. 1))) /\ (dom (k,(ddbi11 /. 1))) is finite countable Element of bool NAT
((len (k,(ddbi11 /. 1))) |-> ((k),(ddbi11 /. 2))) . dbi1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
b12 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b119 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b12,b119*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,b12,b119) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b129 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
b111 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),NAT,(2 -tuples_on (k)),(k,(ddbi11 /. 1)),dbi1) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
b112 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
i2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b112,i2*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,b112,i2) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,i2,b119) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,b112,(k,i2,b119)) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b112,(k,i2,b119)*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
j2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((k),NAT,(2 -tuples_on (k)),(k,D),j2) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
(k,(k,i2,b119)) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
dom (k,(k,i2,b119)) is non empty finite countable V77() Element of bool NAT
<*i2,b119*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
b22 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((k),NAT,(2 -tuples_on (k)),(k,(k,i2,b119)),b22) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
b219 is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
b219 /. 2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k,(b219 /. 2)) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
ddbi21 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
b21 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),ddbi21,b21) is Relation-like NAT -defined (k) -valued Function-like non empty finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
b219 /. 1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
x . j2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
len (k,(b219 /. 2)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
((k),(b219 /. 1)) is Relation-like NAT -defined (k) -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
(len (k,(b219 /. 2))) |-> ((k),(b219 /. 1)) is Relation-like NAT -defined (k) * -valued Function-like finite len (k,(b219 /. 2)) -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () () Element of (len (k,(b219 /. 2))) -tuples_on ((k) *)
(len (k,(b219 /. 2))) -tuples_on ((k) *) is functional non empty FinSequence-membered FinSequenceSet of (k) *
{ b1 where b1 is Relation-like NAT -defined (k) * -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of ((k) *) * : len b1 = len (k,(b219 /. 2)) } is set
Seg (len (k,(b219 /. 2))) is non empty finite len (k,(b219 /. 2)) -element countable Element of bool NAT
(Seg (len (k,(b219 /. 2)))) --> ((k),(b219 /. 1)) is Relation-like non-empty Seg (len (k,(b219 /. 2))) -defined Seg (len (k,(b219 /. 2))) -defined (k) * -valued {((k),(b219 /. 1))} -valued Function-like constant non empty total total quasi_total finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of bool [:(Seg (len (k,(b219 /. 2)))),{((k),(b219 /. 1))}:]
{((k),(b219 /. 1))} is functional non empty trivial finite V40() 1 -element with_common_domain countable with_non-empty_elements non empty-membered set
[:(Seg (len (k,(b219 /. 2)))),{((k),(b219 /. 1))}:] is Relation-like non empty finite countable set
bool [:(Seg (len (k,(b219 /. 2)))),{((k),(b219 /. 1))}:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
(((len (k,(b219 /. 2))) |-> ((k),(b219 /. 1))),(k,(b219 /. 2))) is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () set
dom (x . j2) is finite countable V77() Element of bool NAT
dom ((len (k,(b219 /. 2))) |-> ((k),(b219 /. 1))) is finite len (k,(b219 /. 2)) -element countable V77() Element of bool NAT
dom (k,(b219 /. 2)) is non empty finite countable V77() Element of bool NAT
(dom ((len (k,(b219 /. 2))) |-> ((k),(b219 /. 1)))) /\ (dom (k,(b219 /. 2))) is finite countable Element of bool NAT
(Seg (len (k,(b219 /. 2)))) /\ (dom (k,(b219 /. 2))) is finite countable Element of bool NAT
(dom (k,(b219 /. 2))) /\ (dom (k,(b219 /. 2))) is finite countable Element of bool NAT
(k,(k,i2,b119)) . b22 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
j2 -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x | (j2 -' 1) is Relation-like NAT -defined (3 -tuples_on (k)) * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of (3 -tuples_on (k)) *
((3 -tuples_on (k)),(x | (j2 -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued () FinSequence of NAT
Sum ((3 -tuples_on (k)),(x | (j2 -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum ((3 -tuples_on (k)),(x | (j2 -' 1)))) + b22 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dom ((3 -tuples_on (k)),x) is finite countable V77() Element of bool NAT
((3 -tuples_on (k)),x) . ((Sum ((3 -tuples_on (k)),(x | (j2 -' 1)))) + b22) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
(x . j2) . b22 is set
((len (k,(b219 /. 2))) |-> ((k),(b219 /. 1))) . b22 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
(k,(b219 /. 2)) . b22 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
(((len (k,(b219 /. 2))) |-> ((k),(b219 /. 1))) . b22) ^ ((k,(b219 /. 2)) . b22) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
<*b112*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like () set
<*b112*> ^ <*i2,b119*> is Relation-like NAT -defined Function-like non empty finite 1 + 2 -element countable FinSequence-like FinSubsequence-like () set
1 + 2 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
<*b112,i2,b119*> is Relation-like NAT -defined Function-like non empty finite 3 -element countable FinSequence-like FinSubsequence-like () set
(k,(ddbi11 /. 1)) . dbi1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
<*b119*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like () set
<*b112,i2*> ^ <*b119*> is Relation-like NAT -defined Function-like non empty finite 2 + 1 -element countable FinSequence-like FinSubsequence-like () set
2 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
rng ((3 -tuples_on (k)),x) is functional finite countable FinSequence-membered Element of bool (3 -tuples_on (k))
k2 is set
((3 -tuples_on (k)),x) . k2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
i1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((3 -tuples_on (k)),x) . i1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dbi1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x . j1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
dom (x . j1) is finite countable V77() Element of bool NAT
j1 -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
x | (j1 -' 1) is Relation-like NAT -defined (3 -tuples_on (k)) * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of (3 -tuples_on (k)) *
((3 -tuples_on (k)),(x | (j1 -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued () FinSequence of NAT
Sum ((3 -tuples_on (k)),(x | (j1 -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum ((3 -tuples_on (k)),(x | (j1 -' 1)))) + dbi1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(x . j1) . dbi1 is set
((k),NAT,(2 -tuples_on (k)),(k,D),j1) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
ddbi11 is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
ddbi11 /. 2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k,(ddbi11 /. 2)) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
len (k,(ddbi11 /. 2)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
ddbi11 /. 1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),(ddbi11 /. 1)) is Relation-like NAT -defined (k) -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
(len (k,(ddbi11 /. 2))) |-> ((k),(ddbi11 /. 1)) is Relation-like NAT -defined (k) * -valued Function-like finite len (k,(ddbi11 /. 2)) -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () () Element of (len (k,(ddbi11 /. 2))) -tuples_on ((k) *)
(len (k,(ddbi11 /. 2))) -tuples_on ((k) *) is functional non empty FinSequence-membered FinSequenceSet of (k) *
{ b1 where b1 is Relation-like NAT -defined (k) * -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of ((k) *) * : len b1 = len (k,(ddbi11 /. 2)) } is set
Seg (len (k,(ddbi11 /. 2))) is non empty finite len (k,(ddbi11 /. 2)) -element countable Element of bool NAT
(Seg (len (k,(ddbi11 /. 2)))) --> ((k),(ddbi11 /. 1)) is Relation-like non-empty Seg (len (k,(ddbi11 /. 2))) -defined Seg (len (k,(ddbi11 /. 2))) -defined (k) * -valued {((k),(ddbi11 /. 1))} -valued Function-like constant non empty total total quasi_total finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of bool [:(Seg (len (k,(ddbi11 /. 2)))),{((k),(ddbi11 /. 1))}:]
{((k),(ddbi11 /. 1))} is functional non empty trivial finite V40() 1 -element with_common_domain countable with_non-empty_elements non empty-membered set
[:(Seg (len (k,(ddbi11 /. 2)))),{((k),(ddbi11 /. 1))}:] is Relation-like non empty finite countable set
bool [:(Seg (len (k,(ddbi11 /. 2)))),{((k),(ddbi11 /. 1))}:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
(((len (k,(ddbi11 /. 2))) |-> ((k),(ddbi11 /. 1))),(k,(ddbi11 /. 2))) is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () set
dom (k,(ddbi11 /. 2)) is non empty finite countable V77() Element of bool NAT
dom ((len (k,(ddbi11 /. 2))) |-> ((k),(ddbi11 /. 1))) is finite len (k,(ddbi11 /. 2)) -element countable V77() Element of bool NAT
(dom (k,(ddbi11 /. 2))) /\ (dom ((len (k,(ddbi11 /. 2))) |-> ((k),(ddbi11 /. 1)))) is finite countable Element of bool NAT
(dom (k,(ddbi11 /. 2))) /\ (Seg (len (k,(ddbi11 /. 2)))) is finite countable Element of bool NAT
(dom (k,(ddbi11 /. 2))) /\ (dom (k,(ddbi11 /. 2))) is finite countable Element of bool NAT
((len (k,(ddbi11 /. 2))) |-> ((k),(ddbi11 /. 1))) . dbi1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
b12 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b119 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b12,b119*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,b12,b119) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b129 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
b111 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),NAT,(2 -tuples_on (k)),(k,(ddbi11 /. 2)),dbi1) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
b112 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
i2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b112,i2*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,b112,i2) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,(ddbi11 /. 2)) . dbi1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
<*b12*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like () set
<*b12*> ^ <*b112,i2*> is Relation-like NAT -defined Function-like non empty finite 1 + 2 -element countable FinSequence-like FinSubsequence-like () set
<*b12,b112,i2*> is Relation-like NAT -defined Function-like non empty finite 3 -element countable FinSequence-like FinSubsequence-like () set
(k,b12,b112) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
(k,(k,b12,b112)) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
dom (k,(k,b12,b112)) is non empty finite countable V77() Element of bool NAT
<*b12,b112*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
b21 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((k),NAT,(2 -tuples_on (k)),(k,(k,b12,b112)),b21) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
(k,(k,b12,b112)) . b21 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
(k,(k,b12,b112),i2) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*(k,b12,b112),i2*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
b22 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
((k),NAT,(2 -tuples_on (k)),(k,D),b22) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
b219 is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
b219 /. 1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k,(b219 /. 1)) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
b22 -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M | (b22 -' 1) is Relation-like NAT -defined (3 -tuples_on (k)) * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of (3 -tuples_on (k)) *
((3 -tuples_on (k)),(M | (b22 -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued () FinSequence of NAT
Sum ((3 -tuples_on (k)),(M | (b22 -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum ((3 -tuples_on (k)),(M | (b22 -' 1)))) + b21 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
dbi2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
ddbi21 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),dbi2,ddbi21) is Relation-like NAT -defined (k) -valued Function-like non empty finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
len (k,(b219 /. 1)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
Seg (len (k,(b219 /. 1))) is non empty finite len (k,(b219 /. 1)) -element countable Element of bool NAT
M . b22 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
b219 /. 2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),(b219 /. 2)) is Relation-like NAT -defined (k) -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
(len (k,(b219 /. 1))) |-> ((k),(b219 /. 2)) is Relation-like NAT -defined (k) * -valued Function-like finite len (k,(b219 /. 1)) -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () () Element of (len (k,(b219 /. 1))) -tuples_on ((k) *)
(len (k,(b219 /. 1))) -tuples_on ((k) *) is functional non empty FinSequence-membered FinSequenceSet of (k) *
{ b1 where b1 is Relation-like NAT -defined (k) * -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of ((k) *) * : len b1 = len (k,(b219 /. 1)) } is set
(Seg (len (k,(b219 /. 1)))) --> ((k),(b219 /. 2)) is Relation-like non-empty Seg (len (k,(b219 /. 1))) -defined Seg (len (k,(b219 /. 1))) -defined (k) * -valued {((k),(b219 /. 2))} -valued Function-like constant non empty total total quasi_total finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of bool [:(Seg (len (k,(b219 /. 1)))),{((k),(b219 /. 2))}:]
{((k),(b219 /. 2))} is functional non empty trivial finite V40() 1 -element with_common_domain countable with_non-empty_elements non empty-membered set
[:(Seg (len (k,(b219 /. 1)))),{((k),(b219 /. 2))}:] is Relation-like non empty finite countable set
bool [:(Seg (len (k,(b219 /. 1)))),{((k),(b219 /. 2))}:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
((k,(b219 /. 1)),((len (k,(b219 /. 1))) |-> ((k),(b219 /. 2)))) is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () set
dom (M . b22) is finite countable V77() Element of bool NAT
dom ((len (k,(b219 /. 1))) |-> ((k),(b219 /. 2))) is finite len (k,(b219 /. 1)) -element countable V77() Element of bool NAT
dom (k,(b219 /. 1)) is non empty finite countable V77() Element of bool NAT
(dom ((len (k,(b219 /. 1))) |-> ((k),(b219 /. 2)))) /\ (dom (k,(b219 /. 1))) is finite countable Element of bool NAT
(Seg (len (k,(b219 /. 1)))) /\ (dom (k,(b219 /. 1))) is finite countable Element of bool NAT
(dom (k,(b219 /. 1))) /\ (dom (k,(b219 /. 1))) is finite countable Element of bool NAT
((3 -tuples_on (k)),M) . ((Sum ((3 -tuples_on (k)),(M | (b22 -' 1)))) + b21) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
(M . b22) . b21 is set
(k,(b219 /. 1)) . b21 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
((len (k,(b219 /. 1))) |-> ((k),(b219 /. 2))) . b21 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
((k,(b219 /. 1)) . b21) ^ (((len (k,(b219 /. 1))) |-> ((k),(b219 /. 2))) . b21) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
<*i2*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like () set
<*b12,b112*> ^ <*i2*> is Relation-like NAT -defined Function-like non empty finite 2 + 1 -element countable FinSequence-like FinSubsequence-like () set
k1 is set
dom ((3 -tuples_on (k)),M) is finite countable V77() set
k2 is set
((3 -tuples_on (k)),M) . k1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
((3 -tuples_on (k)),M) . k2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
i1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M . i1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
dom (M . i1) is finite countable V77() Element of bool NAT
i1 -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M | (i1 -' 1) is Relation-like NAT -defined (3 -tuples_on (k)) * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of (3 -tuples_on (k)) *
((3 -tuples_on (k)),(M | (i1 -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued () FinSequence of NAT
Sum ((3 -tuples_on (k)),(M | (i1 -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum ((3 -tuples_on (k)),(M | (i1 -' 1)))) + j1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(M . i1) . j1 is set
((k),NAT,(2 -tuples_on (k)),(k,D),i1) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
dbi1 is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
dbi1 /. 1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k,(dbi1 /. 1)) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
len (k,(dbi1 /. 1)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
dbi1 /. 2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),(dbi1 /. 2)) is Relation-like NAT -defined (k) -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
(len (k,(dbi1 /. 1))) |-> ((k),(dbi1 /. 2)) is Relation-like NAT -defined (k) * -valued Function-like finite len (k,(dbi1 /. 1)) -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () () Element of (len (k,(dbi1 /. 1))) -tuples_on ((k) *)
(len (k,(dbi1 /. 1))) -tuples_on ((k) *) is functional non empty FinSequence-membered FinSequenceSet of (k) *
{ b1 where b1 is Relation-like NAT -defined (k) * -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of ((k) *) * : len b1 = len (k,(dbi1 /. 1)) } is set
Seg (len (k,(dbi1 /. 1))) is non empty finite len (k,(dbi1 /. 1)) -element countable Element of bool NAT
(Seg (len (k,(dbi1 /. 1)))) --> ((k),(dbi1 /. 2)) is Relation-like non-empty Seg (len (k,(dbi1 /. 1))) -defined Seg (len (k,(dbi1 /. 1))) -defined (k) * -valued {((k),(dbi1 /. 2))} -valued Function-like constant non empty total total quasi_total finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of bool [:(Seg (len (k,(dbi1 /. 1)))),{((k),(dbi1 /. 2))}:]
{((k),(dbi1 /. 2))} is functional non empty trivial finite V40() 1 -element with_common_domain countable with_non-empty_elements non empty-membered set
[:(Seg (len (k,(dbi1 /. 1)))),{((k),(dbi1 /. 2))}:] is Relation-like non empty finite countable set
bool [:(Seg (len (k,(dbi1 /. 1)))),{((k),(dbi1 /. 2))}:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
((k,(dbi1 /. 1)),((len (k,(dbi1 /. 1))) |-> ((k),(dbi1 /. 2)))) is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () set
dom (k,(dbi1 /. 1)) is non empty finite countable V77() Element of bool NAT
dom ((len (k,(dbi1 /. 1))) |-> ((k),(dbi1 /. 2))) is finite len (k,(dbi1 /. 1)) -element countable V77() Element of bool NAT
(dom (k,(dbi1 /. 1))) /\ (dom ((len (k,(dbi1 /. 1))) |-> ((k),(dbi1 /. 2)))) is finite countable Element of bool NAT
(dom (k,(dbi1 /. 1))) /\ (Seg (len (k,(dbi1 /. 1)))) is finite countable Element of bool NAT
(dom (k,(dbi1 /. 1))) /\ (dom (k,(dbi1 /. 1))) is finite countable Element of bool NAT
((k),NAT,(2 -tuples_on (k)),(k,(dbi1 /. 1)),j1) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
(k,(dbi1 /. 1)) . j1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
((len (k,(dbi1 /. 1))) |-> ((k),(dbi1 /. 2))) . j1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
b11 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b12 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b11,b12*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,b11,b12) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b119 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
b129 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
b111 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b112 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b111,b112*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,b111,b112) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b12*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like () set
<*b111,b112*> ^ <*b12*> is Relation-like NAT -defined Function-like non empty finite 2 + 1 -element countable FinSequence-like FinSubsequence-like () set
<*b111,b112,b12*> is Relation-like NAT -defined Function-like non empty finite 3 -element countable FinSequence-like FinSubsequence-like () set
i2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
j2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M . i2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
dom (M . i2) is finite countable V77() Element of bool NAT
i2 -' 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
M | (i2 -' 1) is Relation-like NAT -defined (3 -tuples_on (k)) * -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of (3 -tuples_on (k)) *
((3 -tuples_on (k)),(M | (i2 -' 1))) is Relation-like NAT -defined NAT -valued Function-like finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued () FinSequence of NAT
Sum ((3 -tuples_on (k)),(M | (i2 -' 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(Sum ((3 -tuples_on (k)),(M | (i2 -' 1)))) + j2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
(M . i2) . j2 is set
((k),NAT,(2 -tuples_on (k)),(k,D),i2) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
dbi2 is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
dbi2 /. 1 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
(k,(dbi2 /. 1)) is Relation-like NAT -defined 2 -tuples_on (k) -valued Function-like one-to-one non empty finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of 2 -tuples_on (k)
len (k,(dbi2 /. 1)) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal countable V55() real ext-real positive non negative Element of NAT
dbi2 /. 2 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
((k),(dbi2 /. 2)) is Relation-like NAT -defined (k) -valued Function-like constant non empty trivial finite 1 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of (k) *
(len (k,(dbi2 /. 1))) |-> ((k),(dbi2 /. 2)) is Relation-like NAT -defined (k) * -valued Function-like finite len (k,(dbi2 /. 1)) -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () () Element of (len (k,(dbi2 /. 1))) -tuples_on ((k) *)
(len (k,(dbi2 /. 1))) -tuples_on ((k) *) is functional non empty FinSequence-membered FinSequenceSet of (k) *
{ b1 where b1 is Relation-like NAT -defined (k) * -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of ((k) *) * : len b1 = len (k,(dbi2 /. 1)) } is set
Seg (len (k,(dbi2 /. 1))) is non empty finite len (k,(dbi2 /. 1)) -element countable Element of bool NAT
(Seg (len (k,(dbi2 /. 1)))) --> ((k),(dbi2 /. 2)) is Relation-like non-empty Seg (len (k,(dbi2 /. 1))) -defined Seg (len (k,(dbi2 /. 1))) -defined (k) * -valued {((k),(dbi2 /. 2))} -valued Function-like constant non empty total total quasi_total finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of bool [:(Seg (len (k,(dbi2 /. 1)))),{((k),(dbi2 /. 2))}:]
{((k),(dbi2 /. 2))} is functional non empty trivial finite V40() 1 -element with_common_domain countable with_non-empty_elements non empty-membered set
[:(Seg (len (k,(dbi2 /. 1)))),{((k),(dbi2 /. 2))}:] is Relation-like non empty finite countable set
bool [:(Seg (len (k,(dbi2 /. 1)))),{((k),(dbi2 /. 2))}:] is non empty cup-closed diff-closed preBoolean finite V40() countable set
((k,(dbi2 /. 1)),((len (k,(dbi2 /. 1))) |-> ((k),(dbi2 /. 2)))) is Relation-like NAT -defined Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () set
dom (k,(dbi2 /. 1)) is non empty finite countable V77() Element of bool NAT
dom ((len (k,(dbi2 /. 1))) |-> ((k),(dbi2 /. 2))) is finite len (k,(dbi2 /. 1)) -element countable V77() Element of bool NAT
(dom (k,(dbi2 /. 1))) /\ (dom ((len (k,(dbi2 /. 1))) |-> ((k),(dbi2 /. 2)))) is finite countable Element of bool NAT
(dom (k,(dbi2 /. 1))) /\ (Seg (len (k,(dbi2 /. 1)))) is finite countable Element of bool NAT
(dom (k,(dbi2 /. 1))) /\ (dom (k,(dbi2 /. 1))) is finite countable Element of bool NAT
((k),NAT,(2 -tuples_on (k)),(k,(dbi2 /. 1)),j2) is Relation-like NAT -defined (k) -valued Function-like finite 2 -element countable Function-yielding V50() FinSequence-like FinSubsequence-like () Element of 2 -tuples_on (k)
(k,(dbi2 /. 1)) . j2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
((len (k,(dbi2 /. 1))) |-> ((k),(dbi2 /. 2))) . j2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
b21 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b22 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b21,b22*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,b21,b22) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b219 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
b229 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () Element of (k)
b211 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
b212 is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b211,b212*> is Relation-like NAT -defined Function-like non empty finite 2 -element countable FinSequence-like FinSubsequence-like () set
(k,b211,b212) is Relation-like k -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued () set
<*b22*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element countable FinSequence-like FinSubsequence-like () set
<*b211,b212*> ^ <*b22*> is Relation-like NAT -defined Function-like non empty finite 2 + 1 -element countable FinSequence-like FinSubsequence-like () set
<*b211,b212,b22*> is Relation-like NAT -defined Function-like non empty finite 3 -element countable FinSequence-like FinSubsequence-like () set
(k,D) . i2 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
(k,D) . i1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
k is set
D is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
M is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
(k,D,M) is Relation-like k -defined Function-like total complex-valued ext-real-valued real-valued set
((k,D,M)) is set
(D) is set
(M) is set
(D) \/ (M) is set
x is set
(k,D,M) . x is V55() real ext-real set
D . x is V55() real ext-real set
M . x is V55() real ext-real set
(D . x) + (M . x) is V55() real ext-real Element of REAL
D is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
k is non empty set
D -tuples_on k is functional non empty FinSequence-membered FinSequenceSet of k
k * is functional non empty FinSequence-membered FinSequenceSet of k
{ b1 where b1 is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k * : len b1 = D } is set
M is Relation-like NAT -defined D -tuples_on k -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () FinSequence of D -tuples_on k
x is set
dom M is finite countable V77() set
M . x is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
dom M is finite countable V77() Element of bool NAT
k is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
D is non empty set
k -tuples_on D is functional non empty FinSequence-membered FinSequenceSet of D
D * is functional non empty FinSequence-membered FinSequenceSet of D
{ b1 where b1 is Relation-like NAT -defined D -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of D * : len b1 = k } is set
M is Relation-like NAT -defined k -tuples_on D -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of k -tuples_on D
x is set
(D,NAT,(k -tuples_on D),M,x) is Relation-like NAT -defined D -valued Function-like finite k -element countable FinSequence-like FinSubsequence-like () Element of k -tuples_on D
k is epsilon-transitive epsilon-connected ordinal natural finite cardinal countable V55() real ext-real non negative Element of NAT
D is non empty set
k -tuples_on D is functional non empty FinSequence-membered FinSequenceSet of D
D * is functional non empty FinSequence-membered FinSequenceSet of D
{ b1 where b1 is Relation-like NAT -defined D -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of D * : len b1 = k } is set
M is Relation-like NAT -defined k -tuples_on D -valued Function-like finite countable Function-yielding V50() FinSequence-like FinSubsequence-like () () FinSequence of k -tuples_on D
x is set
(D,NAT,(k -tuples_on D),M,x) is Relation-like NAT -defined D -valued Function-like finite k -element countable FinSequence-like FinSubsequence-like () Element of k -tuples_on D